Algebra Examples

$f (x) = 1.25 (38 - x)$

Step 1

Write $f (x) = 1.25 (38 - x)$ as an equation.

$y = 1.25 (38 - x)$

Step 2

Interchange the variables.

$x = 1.25 (38 - y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $1.25 (38 - y) = x$ .

$1.25 (38 - y) = x$

Step 3.2

Divide each term in $1.25 (38 - y) = x$ by $1.25$ and simplify.

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Step 3.2.1

Divide each term in $1.25 (38 - y) = x$ by $1.25$ .

$\frac{1.25 (38 - y)}{1.25} = \frac{x}{1.25}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $1.25$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{1.25 (38 - y)}{1.25} = \frac{x}{1.25}$

Step 3.2.2.1.2

Divide $38 - y$ by $1$ .

$38 - y = \frac{x}{1.25}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply by $1$ .

$38 - y = \frac{1 x}{1.25}$

Step 3.2.3.2

Factor $1.25$ out of $1.25$ .

$38 - y = \frac{1 x}{1.25 (1)}$

Step 3.2.3.3

Separate fractions.

$38 - y = \frac{1}{1.25} \cdot \frac{x}{1}$

Step 3.2.3.4

Divide $1$ by $1.25$ .

$38 - y = 0.8 \frac{x}{1}$

Step 3.2.3.5

Divide $x$ by $1$ .

$38 - y = 0.8 x$

Step 3.3

Subtract $38$ from both sides of the equation.

$- y = 0.8 x - 38$

Step 3.4

Divide each term in $- y = 0.8 x - 38$ by $- 1$ and simplify.

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Step 3.4.1

Divide each term in $- y = 0.8 x - 38$ by $- 1$ .

$\frac{- y}{- 1} = \frac{0.8 x}{- 1} + \frac{- 38}{- 1}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{0.8 x}{- 1} + \frac{- 38}{- 1}$

Step 3.4.2.2

Divide $y$ by $1$ .

$y = \frac{0.8 x}{- 1} + \frac{- 38}{- 1}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Move the negative one from the denominator of $\frac{0.8 x}{- 1}$ .

$y = - 1 \cdot (0.8 x) + \frac{- 38}{- 1}$

Step 3.4.3.1.2

Rewrite $- 1 \cdot (0.8 x)$ as $- (0.8 x)$ .

$y = - (0.8 x) + \frac{- 38}{- 1}$

Step 3.4.3.1.3

Multiply $0.8$ by $- 1$ .

$y = - 0.8 x + \frac{- 38}{- 1}$

Step 3.4.3.1.4

Divide $- 38$ by $- 1$ .

$y = - 0.8 x + 38$

Step 4

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

$f^{- 1} (x) = - 0.8 x + 38$

Step 5

Verify if $f^{- 1} (x) = - 0.8 x + 38$ is the inverse of $f (x) = 1.25 (38 - x)$ .

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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 $f^{- 1} (1.25 (38 - x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (1.25 (38 - x)) = - 0.8 (1.25 (38 - x)) + 38$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the distributive property.

$f^{- 1} (1.25 (38 - x)) = - 0.8 (1.25 \cdot 38 + 1.25 (- x)) + 38$

Step 5.2.3.2

Multiply $1.25$ by $38$ .

$f^{- 1} (1.25 (38 - x)) = - 0.8 (47.5 + 1.25 (- x)) + 38$

Step 5.2.3.3

Multiply $- 1$ by $1.25$ .

$f^{- 1} (1.25 (38 - x)) = - 0.8 (47.5 - 1.25 x) + 38$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (1.25 (38 - x)) = - 0.8 \cdot 47.5 - 0.8 (- 1.25 x) + 38$

Step 5.2.3.5

Multiply $- 0.8$ by $47.5$ .

$f^{- 1} (1.25 (38 - x)) = - 38 - 0.8 (- 1.25 x) + 38$

Step 5.2.3.6

Multiply $- 0.8 (- 1.25 x)$ .

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Step 5.2.3.6.1

Multiply $- 1.25$ by $- 0.8$ .

$f^{- 1} (1.25 (38 - x)) = - 38 + 1 x + 38$

Step 5.2.3.6.2

Multiply $x$ by $1$ .

$f^{- 1} (1.25 (38 - x)) = - 38 + x + 38$

Step 5.2.4

Simplify by adding numbers.

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Step 5.2.4.1

Add $- 38$ and $38$ .

$f^{- 1} (1.25 (38 - x)) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (1.25 (38 - x)) = x$

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 $f (- 0.8 x + 38)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 0.8 x + 38) = 1.25 (38 - (- 0.8 x + 38))$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- 0.8 x + 38) = 1.25 (38 - (- 0.8 x) - 1 \cdot 38)$

Step 5.3.3.2

Multiply $- 0.8$ by $- 1$ .

$f (- 0.8 x + 38) = 1.25 (38 + 0.8 x - 1 \cdot 38)$

Step 5.3.3.3

Multiply $- 1$ by $38$ .

$f (- 0.8 x + 38) = 1.25 (38 + 0.8 x - 38)$

Step 5.3.4

Combine the opposite terms in $38 + 0.8 x - 38$ .

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Step 5.3.4.1

Subtract $38$ from $38$ .

$f (- 0.8 x + 38) = 1.25 (0.8 x + 0)$

Step 5.3.4.2

Add $0.8 x$ and $0$ .

$f (- 0.8 x + 38) = 1.25 (0.8 x)$

Step 5.3.5

Multiply $1.25 (0.8 x)$ .

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Step 5.3.5.1

Multiply $0.8$ by $1.25$ .

$f (- 0.8 x + 38) = 1 x$

Step 5.3.5.2

Multiply $x$ by $1$ .

$f (- 0.8 x + 38) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 0.8 x + 38$ is the inverse of $f (x) = 1.25 (38 - x)$ .

$f^{- 1} (x) = - 0.8 x + 38$