Finite Math Examples

$f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$

Step 1

Write $f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$ as an equation.

$y = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$

Step 2

Interchange the variables.

$x = (9 \sqrt{y} - 5) (7 \sqrt{y} + 3)$

Step 3

Solve for $y$ .

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Rewrite the equation as $(9 \sqrt{y} - 5) (7 \sqrt{y} + 3) = x$ .

$(9 \sqrt{y} - 5) (7 \sqrt{y} + 3) = x$

Solve for $\sqrt{y}$ .

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Simplify $(9 \sqrt{y} - 5) (7 \sqrt{y} + 3)$ .

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Expand $(9 \sqrt{y} - 5) (7 \sqrt{y} + 3)$ using the FOIL Method.

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Apply the distributive property.

$9 \sqrt{y} (7 \sqrt{y} + 3) - 5 (7 \sqrt{y} + 3) = x$

Apply the distributive property.

$9 \sqrt{y} (7 \sqrt{y}) + 9 \sqrt{y} \cdot 3 - 5 (7 \sqrt{y} + 3) = x$

Apply the distributive property.

$9 \sqrt{y} (7 \sqrt{y}) + 9 \sqrt{y} \cdot 3 - 5 (7 \sqrt{y}) - 5 \cdot 3 = x$

Simplify and combine like terms.

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Simplify each term.

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Rewrite using the commutative property of multiplication.

$9 \cdot 7 \sqrt{y} \sqrt{y} + 9 \sqrt{y} \cdot 3 - 5 (7 \sqrt{y}) - 5 \cdot 3 = x$

Multiply $\sqrt{y}$ by $\sqrt{y}$ by adding the exponents.

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Move $\sqrt{y}$ .

$9 \cdot 7 (\sqrt{y} \sqrt{y}) + 9 \sqrt{y} \cdot 3 - 5 (7 \sqrt{y}) - 5 \cdot 3 = x$

Multiply $\sqrt{y}$ by $\sqrt{y}$ .

$9 \cdot 7 {\sqrt{y}}^{2} + 9 \sqrt{y} \cdot 3 - 5 (7 \sqrt{y}) - 5 \cdot 3 = x$

Multiply $9$ by $7$ .

$63 {\sqrt{y}}^{2} + 9 \sqrt{y} \cdot 3 - 5 (7 \sqrt{y}) - 5 \cdot 3 = x$

Multiply $3$ by $9$ .

$63 {\sqrt{y}}^{2} + 27 \sqrt{y} - 5 (7 \sqrt{y}) - 5 \cdot 3 = x$

Multiply $7$ by $- 5$ .

$63 {\sqrt{y}}^{2} + 27 \sqrt{y} - 35 \sqrt{y} - 5 \cdot 3 = x$

Multiply $- 5$ by $3$ .

$63 {\sqrt{y}}^{2} + 27 \sqrt{y} - 35 \sqrt{y} - 15 = x$

Subtract $35 \sqrt{y}$ from $27 \sqrt{y}$ .

$63 {\sqrt{y}}^{2} - 8 \sqrt{y} - 15 = x$

Subtract $x$ from both sides of the equation.

$63 {\sqrt{y}}^{2} - 8 \sqrt{y} - 15 - x = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 63$ , $b = - 8$ , and $c = - 15 - x$ into the quadratic formula and solve for $\sqrt{y}$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (63 \cdot (- 15 - x))}}{2 \cdot 63}$

Simplify.

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Simplify the numerator.

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Raise $- 8$ to the power of $2$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 4 \cdot 63 \cdot (- 15 - x)}}{2 \cdot 63}$

Multiply $- 4$ by $63$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 252 \cdot (- 15 - x)}}{2 \cdot 63}$

Apply the distributive property.

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 252 \cdot - 15 - 252 (- x)}}{2 \cdot 63}$

Multiply $- 252$ by $- 15$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 + 3780 - 252 (- x)}}{2 \cdot 63}$

Multiply $- 1$ by $- 252$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 + 3780 + 252 x}}{2 \cdot 63}$

Add $64$ and $3780$ .

$\sqrt{y} = \frac{8 \pm \sqrt{3844 + 252 x}}{2 \cdot 63}$

Factor $4$ out of $3844 + 252 x$ .

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Factor $4$ out of $3844$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961) + 252 x}}{2 \cdot 63}$

Factor $4$ out of $252 x$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961) + 4 (63 x)}}{2 \cdot 63}$

Factor $4$ out of $4 (961) + 4 (63 x)$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961 + 63 x)}}{2 \cdot 63}$

Rewrite $4 (961 + 63 x)$ as $2^{2} (31^{2} + 63 x)$ .

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Rewrite $4$ as $2^{2}$ .

$\sqrt{y} = \frac{8 \pm \sqrt{2^{2} (961 + 63 x)}}{2 \cdot 63}$

Rewrite $961$ as $31^{2}$ .

$\sqrt{y} = \frac{8 \pm \sqrt{2^{2} (31^{2} + 63 x)}}{2 \cdot 63}$

Pull terms out from under the radical.

$\sqrt{y} = \frac{8 \pm 2 \sqrt{31^{2} + 63 x}}{2 \cdot 63}$

Raise $31$ to the power of $2$ .

$\sqrt{y} = \frac{8 \pm 2 \sqrt{961 + 63 x}}{2 \cdot 63}$

Multiply $2$ by $63$ .

$\sqrt{y} = \frac{8 \pm 2 \sqrt{961 + 63 x}}{126}$

Simplify $\frac{8 \pm 2 \sqrt{961 + 63 x}}{126}$ .

$\sqrt{y} = \frac{4 \pm \sqrt{961 + 63 x}}{63}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 8$ to the power of $2$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 4 \cdot 63 \cdot (- 15 - x)}}{2 \cdot 63}$

Multiply $- 4$ by $63$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 252 \cdot (- 15 - x)}}{2 \cdot 63}$

Apply the distributive property.

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 252 \cdot - 15 - 252 (- x)}}{2 \cdot 63}$

Multiply $- 252$ by $- 15$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 + 3780 - 252 (- x)}}{2 \cdot 63}$

Multiply $- 1$ by $- 252$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 + 3780 + 252 x}}{2 \cdot 63}$

Add $64$ and $3780$ .

$\sqrt{y} = \frac{8 \pm \sqrt{3844 + 252 x}}{2 \cdot 63}$

Factor $4$ out of $3844 + 252 x$ .

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Factor $4$ out of $3844$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961) + 252 x}}{2 \cdot 63}$

Factor $4$ out of $252 x$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961) + 4 (63 x)}}{2 \cdot 63}$

Factor $4$ out of $4 (961) + 4 (63 x)$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961 + 63 x)}}{2 \cdot 63}$

Rewrite $4 (961 + 63 x)$ as $2^{2} (31^{2} + 63 x)$ .

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Rewrite $4$ as $2^{2}$ .

$\sqrt{y} = \frac{8 \pm \sqrt{2^{2} (961 + 63 x)}}{2 \cdot 63}$

Rewrite $961$ as $31^{2}$ .

$\sqrt{y} = \frac{8 \pm \sqrt{2^{2} (31^{2} + 63 x)}}{2 \cdot 63}$

Pull terms out from under the radical.

$\sqrt{y} = \frac{8 \pm 2 \sqrt{31^{2} + 63 x}}{2 \cdot 63}$

Raise $31$ to the power of $2$ .

$\sqrt{y} = \frac{8 \pm 2 \sqrt{961 + 63 x}}{2 \cdot 63}$

Multiply $2$ by $63$ .

$\sqrt{y} = \frac{8 \pm 2 \sqrt{961 + 63 x}}{126}$

Simplify $\frac{8 \pm 2 \sqrt{961 + 63 x}}{126}$ .

$\sqrt{y} = \frac{4 \pm \sqrt{961 + 63 x}}{63}$

Change the $\pm$ to $+$ .

$\sqrt{y} = \frac{4 + \sqrt{961 + 63 x}}{63}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 8$ to the power of $2$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 4 \cdot 63 \cdot (- 15 - x)}}{2 \cdot 63}$

Multiply $- 4$ by $63$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 252 \cdot (- 15 - x)}}{2 \cdot 63}$

Apply the distributive property.

$\sqrt{y} = \frac{8 \pm \sqrt{64 - 252 \cdot - 15 - 252 (- x)}}{2 \cdot 63}$

Multiply $- 252$ by $- 15$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 + 3780 - 252 (- x)}}{2 \cdot 63}$

Multiply $- 1$ by $- 252$ .

$\sqrt{y} = \frac{8 \pm \sqrt{64 + 3780 + 252 x}}{2 \cdot 63}$

Add $64$ and $3780$ .

$\sqrt{y} = \frac{8 \pm \sqrt{3844 + 252 x}}{2 \cdot 63}$

Factor $4$ out of $3844 + 252 x$ .

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Factor $4$ out of $3844$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961) + 252 x}}{2 \cdot 63}$

Factor $4$ out of $252 x$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961) + 4 (63 x)}}{2 \cdot 63}$

Factor $4$ out of $4 (961) + 4 (63 x)$ .

$\sqrt{y} = \frac{8 \pm \sqrt{4 (961 + 63 x)}}{2 \cdot 63}$

Rewrite $4 (961 + 63 x)$ as $2^{2} (31^{2} + 63 x)$ .

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Rewrite $4$ as $2^{2}$ .

$\sqrt{y} = \frac{8 \pm \sqrt{2^{2} (961 + 63 x)}}{2 \cdot 63}$

Rewrite $961$ as $31^{2}$ .

$\sqrt{y} = \frac{8 \pm \sqrt{2^{2} (31^{2} + 63 x)}}{2 \cdot 63}$

Pull terms out from under the radical.

$\sqrt{y} = \frac{8 \pm 2 \sqrt{31^{2} + 63 x}}{2 \cdot 63}$

Raise $31$ to the power of $2$ .

$\sqrt{y} = \frac{8 \pm 2 \sqrt{961 + 63 x}}{2 \cdot 63}$

Multiply $2$ by $63$ .

$\sqrt{y} = \frac{8 \pm 2 \sqrt{961 + 63 x}}{126}$

Simplify $\frac{8 \pm 2 \sqrt{961 + 63 x}}{126}$ .

$\sqrt{y} = \frac{4 \pm \sqrt{961 + 63 x}}{63}$

Change the $\pm$ to $-$ .

$\sqrt{y} = \frac{4 - \sqrt{961 + 63 x}}{63}$

The final answer is the combination of both solutions.

$\sqrt{y} = \frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}$

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

${\sqrt{y}}^{2} = {\frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}}^{2}$

Simplify each side of the equation.

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

${(y^{\frac{1}{2}})}^{2} = {\frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}}^{2}$

Simplify the left side.

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Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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

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

$y^{\frac{1}{2} \cdot 2} = {\frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {\frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}}^{2}$

Rewrite the expression.

$y^{1} = {\frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}}^{2}$

Simplify.

$y = \frac{4 + \sqrt{961 + 63 x}}{63}$

$y = \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$

$y = \frac{4 + \sqrt{961 + 63 x}}{63}$

$y = \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$

$y = {\frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63}}^{2}$

$y = \frac{4 + \sqrt{961 + 63 x}}{63}$

$y = \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$

$y = \frac{4 + \sqrt{961 + 63 x}}{63}$

$y = \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$

Step 4

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

$f^{- 1} (x) = \frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$

Step 5

Verify if $f^{- 1} (x) = \frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$ is the inverse of $f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$ .

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The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$ and $f^{- 1} (x) = \frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$ and compare them.

Find the range of $f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$ .

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The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{961}{63}, - 15]$

Find the domain of $\frac{4 + \sqrt{961 + 63 x}}{63}$ .

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Set the radicand in $\sqrt{961 + 63 x}$ greater than or equal to $0$ to find where the expression is defined.

$961 + 63 x \geq 0$

Solve for $x$ .

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Subtract $961$ from both sides of the inequality.

$63 x \geq - 961$

Divide each term in $63 x \geq - 961$ by $63$ and simplify.

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Divide each term in $63 x \geq - 961$ by $63$ .

$\frac{63 x}{63} \geq \frac{- 961}{63}$

Simplify the left side.

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Cancel the common factor of $63$ .

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Cancel the common factor.

$\frac{63 x}{63} \geq \frac{- 961}{63}$

Divide $x$ by $1$ .

$x \geq \frac{- 961}{63}$

Simplify the right side.

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Move the negative in front of the fraction.

$x \geq - \frac{961}{63}$

The domain is all values of $x$ that make the expression defined.

$[- \frac{961}{63}, \infty)$

Since the domain of $f^{- 1} (x) = \frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$ is not equal to the range of $f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$ , then $f^{- 1} (x) = \frac{4 + \sqrt{961 + 63 x}}{63}, \frac{4 - \sqrt{961 + 63 x}}{63^{2}}$ is not an inverse of $f (x) = (9 \sqrt{x} - 5) (7 \sqrt{x} + 3)$ .

There is no inverse

Step 6