Find the Inverse h(x)=5/2x+4

$h (x) = \frac{5}{2} x + 4$

Step 1

Write $h (x) = \frac{5}{2} x + 4$ as an equation.

$y = \frac{5}{2} x + 4$

Step 2

Interchange the variables.

$x = \frac{5}{2} y + 4$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{5}{2} y + 4 = x$ .

$\frac{5}{2} y + 4 = x$

Combine $\frac{5}{2}$ and $y$ .

$\frac{5 y}{2} + 4 = x$

Subtract $4$ from both sides of the equation.

$\frac{5 y}{2} = x - 4$

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} \cdot \frac{5 y}{2} = \frac{2}{5} (x - 4)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{5} \cdot \frac{5 y}{2}$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{5} \cdot \frac{5 y}{2} = \frac{2}{5} (x - 4)$

Rewrite the expression.

$\frac{1}{5} (5 y) = \frac{2}{5} (x - 4)$

Cancel the common factor of $5$ .

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) = \frac{2}{5} (x - 4)$

Cancel the common factor.

$\frac{1}{5} (5 y) = \frac{2}{5} (x - 4)$

Rewrite the expression.

$y = \frac{2}{5} (x - 4)$

Simplify the right side.

Simplify $\frac{2}{5} (x - 4)$ .

Apply the distributive property.

$y = \frac{2}{5} x + \frac{2}{5} \cdot - 4$

Combine $\frac{2}{5}$ and $x$ .

$y = \frac{2 x}{5} + \frac{2}{5} \cdot - 4$

Multiply $\frac{2}{5} \cdot - 4$ .

Combine $\frac{2}{5}$ and $- 4$ .

$y = \frac{2 x}{5} + \frac{2 \cdot - 4}{5}$

Multiply $2$ by $- 4$ .

$y = \frac{2 x}{5} + \frac{- 8}{5}$

Move the negative in front of the fraction.

$y = \frac{2 x}{5} - \frac{8}{5}$

Step 4

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

$h^{- 1} (x) = \frac{2 x}{5} - \frac{8}{5}$

Step 5

Verify if $h^{- 1} (x) = \frac{2 x}{5} - \frac{8}{5}$ is the inverse of $h (x) = \frac{5}{2} x + 4$ .

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

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

Set up the composite result function.

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

Evaluate $h^{- 1} (\frac{5}{2} x + 4)$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (\frac{5}{2} x + 4) = \frac{2 (\frac{5}{2} x + 4)}{5} - \frac{8}{5}$

Combine the numerators over the common denominator.

$h^{- 1} (\frac{5}{2} x + 4) = \frac{2 (\frac{5}{2} x + 4) - 8}{5}$

Simplify each term.

Combine $\frac{5}{2}$ and $x$ .

$h^{- 1} (\frac{5}{2} x + 4) = \frac{2 (\frac{5 x}{2} + 4) - 8}{5}$

Apply the distributive property.

$h^{- 1} (\frac{5}{2} x + 4) = \frac{2 (\frac{5 x}{2}) + 2 \cdot 4 - 8}{5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$h^{- 1} (\frac{5}{2} x + 4) = \frac{2 (\frac{5 x}{2}) + 2 \cdot 4 - 8}{5}$

Rewrite the expression.

$h^{- 1} (\frac{5}{2} x + 4) = \frac{5 x + 2 \cdot 4 - 8}{5}$

Multiply $2$ by $4$ .

$h^{- 1} (\frac{5}{2} x + 4) = \frac{5 x + 8 - 8}{5}$

Simplify terms.

Combine the opposite terms in $5 x + 8 - 8$ .

Subtract $8$ from $8$ .

$h^{- 1} (\frac{5}{2} x + 4) = \frac{5 x + 0}{5}$

Add $5 x$ and $0$ .

$h^{- 1} (\frac{5}{2} x + 4) = \frac{5 x}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$h^{- 1} (\frac{5}{2} x + 4) = \frac{5 x}{5}$

Divide $x$ by $1$ .

$h^{- 1} (\frac{5}{2} x + 4) = x$

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

Set up the composite result function.

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

Evaluate $h (\frac{2 x}{5} - \frac{8}{5})$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (\frac{2 x}{5} - \frac{8}{5}) = \frac{5}{2} \cdot (\frac{2 x}{5} - \frac{8}{5}) + 4$

Simplify each term.

Apply the distributive property.

$h (\frac{2 x}{5} - \frac{8}{5}) = \frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot (- \frac{8}{5}) + 4$

Cancel the common factor of $5$ .

Cancel the common factor.

$h (\frac{2 x}{5} - \frac{8}{5}) = \frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot (- \frac{8}{5}) + 4$

Rewrite the expression.

$h (\frac{2 x}{5} - \frac{8}{5}) = \frac{1}{2} \cdot (2 x) + \frac{5}{2} \cdot (- \frac{8}{5}) + 4$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$h (\frac{2 x}{5} - \frac{8}{5}) = \frac{1}{2} \cdot (2 (x)) + \frac{5}{2} \cdot (- \frac{8}{5}) + 4$

Cancel the common factor.

$h (\frac{2 x}{5} - \frac{8}{5}) = \frac{1}{2} \cdot (2 x) + \frac{5}{2} \cdot (- \frac{8}{5}) + 4$

Rewrite the expression.

$h (\frac{2 x}{5} - \frac{8}{5}) = x + \frac{5}{2} \cdot (- \frac{8}{5}) + 4$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{8}{5}$ into the numerator.

$h (\frac{2 x}{5} - \frac{8}{5}) = x + \frac{5}{2} \cdot \frac{- 8}{5} + 4$

Cancel the common factor.

$h (\frac{2 x}{5} - \frac{8}{5}) = x + \frac{5}{2} \cdot \frac{- 8}{5} + 4$

Rewrite the expression.

$h (\frac{2 x}{5} - \frac{8}{5}) = x + \frac{1}{2} \cdot - 8 + 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 8$ .

$h (\frac{2 x}{5} - \frac{8}{5}) = x + \frac{1}{2} \cdot (2 (- 4)) + 4$

Cancel the common factor.

$h (\frac{2 x}{5} - \frac{8}{5}) = x + \frac{1}{2} \cdot (2 \cdot - 4) + 4$

Rewrite the expression.

$h (\frac{2 x}{5} - \frac{8}{5}) = x - 4 + 4$

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

Add $- 4$ and $4$ .

$h (\frac{2 x}{5} - \frac{8}{5}) = x + 0$

Add $x$ and $0$ .

$h (\frac{2 x}{5} - \frac{8}{5}) = x$

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = \frac{2 x}{5} - \frac{8}{5}$ is the inverse of $h (x) = \frac{5}{2} x + 4$ .

$h^{- 1} (x) = \frac{2 x}{5} - \frac{8}{5}$

Find the Inverse h(x)=5/2x+4

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