Finite Math Examples

$f (x) = - 7^{2} + 3 x + 3$

Step 1

Write $f (x) = - 7^{2} + 3 x + 3$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 7^{2} + 3 y + 3 = x$ .

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

Step 3.2

Simplify $- 7^{2} + 3 y + 3$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $7$ to the power of $2$ .

$- 1 \cdot 49 + 3 y + 3 = x$

Step 3.2.1.2

Multiply $- 1$ by $49$ .

$- 49 + 3 y + 3 = x$

Step 3.2.2

Add $- 49$ and $3$ .

$3 y - 46 = x$

Step 3.3

Add $46$ to both sides of the equation.

$3 y = x + 46$

Step 3.4

Divide each term in $3 y = x + 46$ by $3$ and simplify.

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Step 3.4.1

Divide each term in $3 y = x + 46$ by $3$ .

$\frac{3 y}{3} = \frac{x}{3} + \frac{46}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x}{3} + \frac{46}{3}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{3} + \frac{46}{3}$

Step 4

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

$f^{- 1} (x) = \frac{x}{3} + \frac{46}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{3} + \frac{46}{3}$ is the inverse of $f (x) = - 7^{2} + 3 x + 3$ .

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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 $f^{- 1} (- 7^{2} + 3 x + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 7^{2} + 3 x + 3) = \frac{- 7^{2} + 3 x + 3}{3} + \frac{46}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (- 7^{2} + 3 x + 3) = \frac{- 7^{2} + 3 x + 3 + 46}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Raise $7$ to the power of $2$ .

$f^{- 1} (- 7^{2} + 3 x + 3) = \frac{- 1 \cdot 49 + 3 x + 3 + 46}{3}$

Step 5.2.4.2

Multiply $- 1$ by $49$ .

$f^{- 1} (- 7^{2} + 3 x + 3) = \frac{- 49 + 3 x + 3 + 46}{3}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Add $- 49$ and $3$ .

$f^{- 1} (- 7^{2} + 3 x + 3) = \frac{3 x - 46 + 46}{3}$

Step 5.2.5.2

Combine the opposite terms in $3 x - 46 + 46$ .

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Step 5.2.5.2.1

Add $- 46$ and $46$ .

$f^{- 1} (- 7^{2} + 3 x + 3) = \frac{3 x + 0}{3}$

Step 5.2.5.2.2

Add $3 x$ and $0$ .

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

Step 5.2.5.3

Cancel the common factor of $3$ .

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Step 5.2.5.3.1

Cancel the common factor.

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

Step 5.2.5.3.2

Divide $x$ by $1$ .

$f^{- 1} (- 7^{2} + 3 x + 3) = x$

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 $f (\frac{x}{3} + \frac{46}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{3} + \frac{46}{3}) = - 7^{2} + 3 (\frac{x}{3} + \frac{46}{3}) + 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Raise $7$ to the power of $2$ .

$f (\frac{x}{3} + \frac{46}{3}) = - 1 \cdot 49 + 3 (\frac{x}{3} + \frac{46}{3}) + 3$

Step 5.3.3.2

Multiply $- 1$ by $49$ .

$f (\frac{x}{3} + \frac{46}{3}) = - 49 + 3 (\frac{x}{3} + \frac{46}{3}) + 3$

Step 5.3.3.3

Apply the distributive property.

$f (\frac{x}{3} + \frac{46}{3}) = - 49 + 3 (\frac{x}{3}) + 3 (\frac{46}{3}) + 3$

Step 5.3.3.4

Cancel the common factor of $3$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (\frac{x}{3} + \frac{46}{3}) = - 49 + 3 (\frac{x}{3}) + 3 (\frac{46}{3}) + 3$

Step 5.3.3.4.2

Rewrite the expression.

$f (\frac{x}{3} + \frac{46}{3}) = - 49 + x + 3 (\frac{46}{3}) + 3$

Step 5.3.3.5

Cancel the common factor of $3$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{x}{3} + \frac{46}{3}) = - 49 + x + 3 (\frac{46}{3}) + 3$

Step 5.3.3.5.2

Rewrite the expression.

$f (\frac{x}{3} + \frac{46}{3}) = - 49 + x + 46 + 3$

Step 5.3.4

Simplify by adding terms.

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Step 5.3.4.1

Add $- 49$ and $46$ .

$f (\frac{x}{3} + \frac{46}{3}) = x - 3 + 3$

Step 5.3.4.2

Combine the opposite terms in $x - 3 + 3$ .

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Step 5.3.4.2.1

Add $- 3$ and $3$ .

$f (\frac{x}{3} + \frac{46}{3}) = x + 0$

Step 5.3.4.2.2

Add $x$ and $0$ .

$f (\frac{x}{3} + \frac{46}{3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{3} + \frac{46}{3}$ is the inverse of $f (x) = - 7^{2} + 3 x + 3$ .

$f^{- 1} (x) = \frac{x}{3} + \frac{46}{3}$