Algebra Examples

$f (x) = - 3 \sqrt{\frac{4 x - 7}{3}}$

Step 1

Write $f (x) = - 3 \sqrt{\frac{4 x - 7}{3}}$ as an equation.

$y = - 3 \sqrt{\frac{4 x - 7}{3}}$

Step 2

Interchange the variables.

$x = - 3 \sqrt{\frac{4 y - 7}{3}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 3 \sqrt{\frac{4 y - 7}{3}} = x$ .

$- 3 \sqrt{\frac{4 y - 7}{3}} = x$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 3 \sqrt{\frac{4 y - 7}{3}})}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{4 y - 7}{3}}$ as ${(\frac{4 y - 7}{3})}^{\frac{1}{2}}$ .

${(- 3 {(\frac{4 y - 7}{3})}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${(- 3 {(\frac{4 y - 7}{3})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Apply the product rule to $\frac{4 y - 7}{3}$ .

${(- 3 \frac{{(4 y - 7)}^{\frac{1}{2}}}{3^{\frac{1}{2}}})}^{2} = x^{2}$

Step 3.3.2.1.2

Combine $- 3$ and $\frac{{(4 y - 7)}^{\frac{1}{2}}}{3^{\frac{1}{2}}}$ .

${(\frac{- 3 {(4 y - 7)}^{\frac{1}{2}}}{3^{\frac{1}{2}}})}^{2} = x^{2}$

Step 3.3.2.1.3

Move the negative in front of the fraction.

${(- \frac{(3) {(4 y - 7)}^{\frac{1}{2}}}{3^{\frac{1}{2}}})}^{2} = x^{2}$

Step 3.3.2.1.4

Move $3^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(- ((3) {(4 y - 7)}^{\frac{1}{2}} \cdot 3^{- \frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.5

Multiply $3$ by $3^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.3.2.1.5.1

Move $3^{- \frac{1}{2}}$ .

${(- (3^{- \frac{1}{2}} \cdot 3 {(4 y - 7)}^{\frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.5.2

Multiply $3^{- \frac{1}{2}}$ by $3$ .

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Step 3.3.2.1.5.2.1

Raise $3$ to the power of $1$ .

${(- (3^{- \frac{1}{2}} \cdot 3^{1} {(4 y - 7)}^{\frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(- (3^{- \frac{1}{2} + 1} {(4 y - 7)}^{\frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.5.3

Write $1$ as a fraction with a common denominator.

${(- (3^{- \frac{1}{2} + \frac{2}{2}} {(4 y - 7)}^{\frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.5.4

Combine the numerators over the common denominator.

${(- (3^{\frac{- 1 + 2}{2}} {(4 y - 7)}^{\frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.5.5

Add $- 1$ and $2$ .

${(- (3^{\frac{1}{2}} {(4 y - 7)}^{\frac{1}{2}}))}^{2} = x^{2}$

Step 3.3.2.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.6.1

Apply the product rule to $- 3^{\frac{1}{2}} {(4 y - 7)}^{\frac{1}{2}}$ .

${(- 3^{\frac{1}{2}})}^{2} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.6.2

Apply the product rule to $- 3^{\frac{1}{2}}$ .

${(- 1)}^{2} {(3^{\frac{1}{2}})}^{2} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.7

Raise $- 1$ to the power of $2$ .

$1 {(3^{\frac{1}{2}})}^{2} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.8

Multiply ${(3^{\frac{1}{2}})}^{2}$ by $1$ .

${(3^{\frac{1}{2}})}^{2} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.9

Multiply the exponents in ${(3^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{\frac{1}{2} \cdot 2} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.9.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.9.2.1

Cancel the common factor.

$3^{\frac{1}{2} \cdot 2} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.9.2.2

Rewrite the expression.

$3^{1} {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.10

Evaluate the exponent.

$3 {({(4 y - 7)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2.1.11

Multiply the exponents in ${({(4 y - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 {(4 y - 7)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.11.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.11.2.1

Cancel the common factor.

$3 {(4 y - 7)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.11.2.2

Rewrite the expression.

$3 {(4 y - 7)}^{1} = x^{2}$

Step 3.3.2.1.12

Simplify.

$3 (4 y - 7) = x^{2}$

Step 3.3.2.1.13

Apply the distributive property.

$3 (4 y) + 3 \cdot - 7 = x^{2}$

Step 3.3.2.1.14

Multiply.

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Step 3.3.2.1.14.1

Multiply $4$ by $3$ .

$12 y + 3 \cdot - 7 = x^{2}$

Step 3.3.2.1.14.2

Multiply $3$ by $- 7$ .

$12 y - 21 = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $21$ to both sides of the equation.

$12 y = x^{2} + 21$

Step 3.4.2

Divide each term in $12 y = x^{2} + 21$ by $12$ and simplify.

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Step 3.4.2.1

Divide each term in $12 y = x^{2} + 21$ by $12$ .

$\frac{12 y}{12} = \frac{x^{2}}{12} + \frac{21}{12}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $12$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{x^{2}}{12} + \frac{21}{12}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{12} + \frac{21}{12}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Cancel the common factor of $21$ and $12$ .

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Step 3.4.2.3.1.1

Factor $3$ out of $21$ .

$y = \frac{x^{2}}{12} + \frac{3 (7)}{12}$

Step 3.4.2.3.1.2

Cancel the common factors.

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Step 3.4.2.3.1.2.1

Factor $3$ out of $12$ .

$y = \frac{x^{2}}{12} + \frac{3 \cdot 7}{3 \cdot 4}$

Step 3.4.2.3.1.2.2

Cancel the common factor.

$y = \frac{x^{2}}{12} + \frac{3 \cdot 7}{3 \cdot 4}$

Step 3.4.2.3.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{12} + \frac{7}{4}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{x^{2}}{12} + \frac{7}{4}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{12} + \frac{7}{4}$ is the inverse of $f (x) = - 3 \sqrt{\frac{4 x - 7}{3}}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{{(- 3 \sqrt{\frac{4 x - 7}{3}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Apply the product rule to $- 3 \sqrt{\frac{4 x - 7}{3}}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{{(- 3)}^{2} {\sqrt{\frac{4 x - 7}{3}}}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.2

Raise $- 3$ to the power of $2$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {\sqrt{\frac{4 x - 7}{3}}}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.3

Rewrite $\sqrt{\frac{4 x - 7}{3}}$ as $\frac{\sqrt{4 x - 7}}{\sqrt{3}}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7}}{\sqrt{3}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.4

Multiply $\frac{\sqrt{4 x - 7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5

Combine and simplify the denominator.

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Step 5.2.3.1.5.1

Multiply $\frac{\sqrt{4 x - 7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.5

Add $1$ and $1$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{{\sqrt{3}}^{2}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.3.1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{3^{\frac{2}{2}}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.6.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.5.6.4.1

Cancel the common factor.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{3^{\frac{2}{2}}})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.6.4.2

Rewrite the expression.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{3})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.5.6.5

Evaluate the exponent.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{4 x - 7} \sqrt{3}}{3})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.6

Combine using the product rule for radicals.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 {(\frac{\sqrt{(4 x - 7) \cdot 3}}{3})}^{2}}{12} + \frac{7}{4}$

Step 5.2.3.1.7

Apply the product rule to $\frac{\sqrt{(4 x - 7) \cdot 3}}{3}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{{\sqrt{(4 x - 7) \cdot 3}}^{2}}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8

Simplify the numerator.

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Step 5.2.3.1.8.1

Rewrite ${\sqrt{(4 x - 7) \cdot 3}}^{2}$ as $(4 x - 7) \cdot 3$ .

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Step 5.2.3.1.8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 x - 7) \cdot 3}$ as ${((4 x - 7) \cdot 3)}^{\frac{1}{2}}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{{({((4 x - 7) \cdot 3)}^{\frac{1}{2}})}^{2}}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{{((4 x - 7) \cdot 3)}^{\frac{1}{2} \cdot 2}}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.1.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{{((4 x - 7) \cdot 3)}^{\frac{2}{2}}}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.1.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.8.1.4.1

Cancel the common factor.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{{((4 x - 7) \cdot 3)}^{\frac{2}{2}}}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.1.4.2

Rewrite the expression.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{(4 x - 7) \cdot 3}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.1.5

Simplify.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{(4 x - 7) \cdot 3}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.2

Apply the distributive property.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{4 x \cdot 3 - 7 \cdot 3}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.3

Multiply $3$ by $4$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{12 x - 7 \cdot 3}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.4

Multiply $- 7$ by $3$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{12 x - 21}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.5

Factor $3$ out of $12 x - 21$ .

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Step 5.2.3.1.8.5.1

Factor $3$ out of $12 x$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{3 (4 x) - 21}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.5.2

Factor $3$ out of $- 21$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{3 (4 x) + 3 (- 7)}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.8.5.3

Factor $3$ out of $3 (4 x) + 3 (- 7)$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{3 (4 x - 7)}{3^{2}})}{12} + \frac{7}{4}$

Step 5.2.3.1.9

Raise $3$ to the power of $2$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{3 (4 x - 7)}{9})}{12} + \frac{7}{4}$

Step 5.2.3.1.10

Cancel the common factors.

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Step 5.2.3.1.10.1

Factor $3$ out of $9$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{3 (4 x - 7)}{3 \cdot 3})}{12} + \frac{7}{4}$

Step 5.2.3.1.10.2

Cancel the common factor.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{3 (4 x - 7)}{3 \cdot 3})}{12} + \frac{7}{4}$

Step 5.2.3.1.10.3

Rewrite the expression.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{9 (\frac{4 x - 7}{3})}{12} + \frac{7}{4}$

Step 5.2.3.2

Combine $9$ and $\frac{4 x - 7}{3}$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{\frac{9 (4 x - 7)}{3}}{12} + \frac{7}{4}$

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.3.1

Reduce the expression $\frac{9 (4 x - 7)}{3}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $3$ out of $9 (4 x - 7)$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{\frac{3 (3 (4 x - 7))}{3}}{12} + \frac{7}{4}$

Step 5.2.3.3.1.2

Factor $3$ out of $3$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{\frac{3 (3 (4 x - 7))}{3 (1)}}{12} + \frac{7}{4}$

Step 5.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{\frac{3 (3 (4 x - 7))}{3 \cdot 1}}{12} + \frac{7}{4}$

Step 5.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{\frac{3 (4 x - 7)}{1}}{12} + \frac{7}{4}$

Step 5.2.3.3.2

Divide $3 (4 x - 7)$ by $1$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{3 (4 x - 7)}{12} + \frac{7}{4}$

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $3$ out of $12$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{3 (4 x - 7)}{3 \cdot 4} + \frac{7}{4}$

Step 5.2.3.4.2

Cancel the common factor.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{3 (4 x - 7)}{3 \cdot 4} + \frac{7}{4}$

Step 5.2.3.4.3

Rewrite the expression.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{4 x - 7}{4} + \frac{7}{4}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{4 x - 7 + 7}{4}$

Step 5.2.4.2

Combine the opposite terms in $4 x - 7 + 7$ .

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Step 5.2.4.2.1

Add $- 7$ and $7$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{4 x + 0}{4}$

Step 5.2.4.2.2

Add $4 x$ and $0$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{4 x}{4}$

Step 5.2.4.3

Cancel the common factor of $4$ .

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Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = \frac{4 x}{4}$

Step 5.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (- 3 \sqrt{\frac{4 x - 7}{3}}) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{x^{2}}{12} + \frac{7}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{4 (\frac{x^{2}}{12} + \frac{7}{4}) - 7}{3}}$

Step 5.3.3

Apply the distributive property.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{4 (\frac{x^{2}}{12}) + 4 (\frac{7}{4}) - 7}{3}}$

Step 5.3.4

Cancel the common factor of $4$ .

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Step 5.3.4.1

Factor $4$ out of $12$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{4 (\frac{x^{2}}{4 (3)}) + 4 (\frac{7}{4}) - 7}{3}}$

Step 5.3.4.2

Cancel the common factor.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{4 (\frac{x^{2}}{4 \cdot 3}) + 4 (\frac{7}{4}) - 7}{3}}$

Step 5.3.4.3

Rewrite the expression.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{\frac{x^{2}}{3} + 4 (\frac{7}{4}) - 7}{3}}$

Step 5.3.5

Cancel the common factor of $4$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{\frac{x^{2}}{3} + 4 (\frac{7}{4}) - 7}{3}}$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{\frac{x^{2}}{3} + 7 - 7}{3}}$

Step 5.3.6

Simplify by subtracting numbers.

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Step 5.3.6.1

Subtract $7$ from $7$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{\frac{x^{2}}{3} + 0}{3}}$

Step 5.3.6.2

Add $\frac{x^{2}}{3}$ and $0$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{\frac{x^{2}}{3}}{3}}$

Step 5.3.7

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{x^{2}}{3} \cdot \frac{1}{3}}$

Step 5.3.8

Combine.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{x^{2} \cdot 1}{3 \cdot 3}}$

Step 5.3.9

Write the expression using exponents.

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Step 5.3.9.1

Multiply $x^{2}$ by $1$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{x^{2}}{3 \cdot 3}}$

Step 5.3.9.2

Multiply $3$ by $3$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{x^{2}}{9}}$

Step 5.3.9.3

Rewrite $9$ as $3^{2}$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{\frac{x^{2}}{3^{2}}}$

Step 5.3.10

Rewrite $\frac{x^{2}}{3^{2}}$ as ${(\frac{x}{3})}^{2}$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \sqrt{{(\frac{x}{3})}^{2}}$

Step 5.3.11

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 3 \frac{x}{3}$

Step 5.3.12

Cancel the common factor of $3$ .

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Step 5.3.12.1

Factor $3$ out of $- 3$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = 3 (- 1) (\frac{x}{3})$

Step 5.3.12.2

Cancel the common factor.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = 3 \cdot (- 1 \frac{x}{3})$

Step 5.3.12.3

Rewrite the expression.

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - 1 x$

Step 5.3.13

Rewrite $- 1 x$ as $- x$ .

$f (\frac{x^{2}}{12} + \frac{7}{4}) = - x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{12} + \frac{7}{4}$ is the inverse of $f (x) = - 3 \sqrt{\frac{4 x - 7}{3}}$ .

$f^{- 1} (x) = \frac{x^{2}}{12} + \frac{7}{4}$