Graph f^-1(x)=(2x)/3-8/3

$f^{- 1} (x) = \frac{2 x}{3} - \frac{8}{3}$

Step 1

Rewrite the function as an equation.

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Reorder terms.

$y = \frac{2}{3} x - \frac{8}{3}$

Step 3

Use the slope-intercept form to find the slope and y-intercept.

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = \frac{2}{3}$

$b = - \frac{8}{3}$

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $\frac{2}{3}$

y-intercept: $(0, - \frac{8}{3})$

Slope: $\frac{2}{3}$

y-intercept: $(0, - \frac{8}{3})$

Step 4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

Reorder terms.

$y = \frac{2}{3} x - \frac{8}{3}$

Find the x-intercept.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{2}{3} x - \frac{8}{3}$

Solve the equation.

Rewrite the equation as $\frac{2}{3} x - \frac{8}{3} = 0$ .

$\frac{2}{3} x - \frac{8}{3} = 0$

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} - \frac{8}{3} = 0$

Add $\frac{8}{3}$ to both sides of the equation.

$\frac{2 x}{3} = \frac{8}{3}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x = 8$

Divide each term in $2 x = 8$ by $2$ and simplify.

Divide each term in $2 x = 8$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2}$

Divide $x$ by $1$ .

$x = \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$x = 4$

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

Find the y-intercept.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{2}{3} \cdot (0) - \frac{8}{3}$

Solve the equation.

Multiply $\frac{2}{3}$ by $0$ .

$y = \frac{2}{3} \cdot 0 - \frac{8}{3}$

Remove parentheses.

$y = \frac{2}{3} \cdot (0) - \frac{8}{3}$

Simplify $\frac{2}{3} \cdot (0) - \frac{8}{3}$ .

Multiply $\frac{2}{3}$ by $0$ .

$y = 0 - \frac{8}{3}$

Subtract $\frac{8}{3}$ from $0$ .

$y = - \frac{8}{3}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{8}{3})$

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ 0 & - \frac{8}{3} \\ 4 & 0 \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope: $\frac{2}{3}$

y-intercept: $(0, - \frac{8}{3})$

$\begin{array}{c} x & y \\ 0 & - \frac{8}{3} \\ 4 & 0 \end{array}$

Step 6

Graph f^-1(x)=(2x)/3-8/3

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