1. We are asked to find the inverse of each given function and then graph both the function and its inverse. 2. To find the inverse of a function $f(x)$, we swap $x$ and $y$ in the equation $y=f(x)$ and then solve for $y$. --- **13) Find the inverse of** $g(x) = \frac{7x + 18}{2}$ Step 1: Write $y = \frac{7x + 18}{2}$. Step 2: Swap $x$ and $y$: $$x = \frac{7y + 18}{2}$$ Step 3: Multiply both sides by 2: $$2x = 7y + 18$$ Step 4: Isolate $y$: $$7y = 2x - 18$$ Step 5: Divide both sides by 7: $$y = \frac{2x - 18}{7}$$ So, the inverse is $$g^{-1}(x) = \frac{2x - 18}{7}$$. --- **14) Find the inverse of** $f(x) = x + 3$ Step 1: Write $y = x + 3$. Step 2: Swap $x$ and $y$: $$x = y + 3$$ Step 3: Isolate $y$: $$y = x - 3$$ So, the inverse is $$f^{-1}(x) = x - 3$$. --- **15) Find the inverse of** $f(x) = -x + 3$ Step 1: Write $y = -x + 3$. Step 2: Swap $x$ and $y$: $$x = -y + 3$$ Step 3: Isolate $y$: $$-y = x - 3$$ Step 4: Multiply both sides by $-1$: $$y = -x + 3$$ So, the inverse is $$f^{-1}(x) = -x + 3$$. --- **16) Find the inverse of** $f(x) = 4x$ Step 1: Write $y = 4x$. Step 2: Swap $x$ and $y$: $$x = 4y$$ Step 3: Divide both sides by 4: $$y = \frac{x}{4}$$ So, the inverse is $$f^{-1}(x) = \frac{x}{4}$$. --- **17) Find the inverse of** $f(x) = -1 - \frac{1}{5}x$ Step 1: Write $y = -1 - \frac{1}{5}x$. Step 2: Swap $x$ and $y$: $$x = -1 - \frac{1}{5}y$$ Step 3: Add 1 to both sides: $$x + 1 = -\frac{1}{5}y$$ Step 4: Multiply both sides by $-5$: $$-5(x + 1) = y$$ So, the inverse is $$f^{-1}(x) = -5(x + 1) = -5x - 5$$. --- **18) Find the inverse of** $g(x) = \frac{1}{x - 1}$ Step 1: Write $y = \frac{1}{x - 1}$. Step 2: Swap $x$ and $y$: $$x = \frac{1}{y - 1}$$ Step 3: Multiply both sides by $y - 1$: $$x(y - 1) = 1$$ Step 4: Distribute $x$: $$xy - x = 1$$ Step 5: Add $x$ to both sides: $$xy = x + 1$$ Step 6: Divide both sides by $x$: $$y = \frac{x + 1}{x}$$ So, the inverse is $$g^{-1}(x) = \frac{x + 1}{x}$$. --- **19) Find the inverse of** $f(x) = -2x^3 + 1$ Step 1: Write $y = -2x^3 + 1$. Step 2: Swap $x$ and $y$: $$x = -2y^3 + 1$$ Step 3: Subtract 1 from both sides: $$x - 1 = -2y^3$$ Step 4: Divide both sides by $-2$: $$\frac{x - 1}{-2} = y^3$$ Step 5: Take the cube root of both sides: $$y = \sqrt[3]{\frac{x - 1}{-2}} = \sqrt[3]{-\frac{x - 1}{2}}$$ So, the inverse is $$f^{-1}(x) = \sqrt[3]{-\frac{x - 1}{2}}$$. --- **20) Find the inverse of** $g(x) = \frac{-x - 5}{3}$ Step 1: Write $y = \frac{-x - 5}{3}$. Step 2: Swap $x$ and $y$: $$x = \frac{-y - 5}{3}$$ Step 3: Multiply both sides by 3: $$3x = -y - 5$$ Step 4: Add 5 to both sides: $$3x + 5 = -y$$ Step 5: Multiply both sides by $-1$: $$y = -3x - 5$$ So, the inverse is $$g^{-1}(x) = -3x - 5$$. --- **Summary of inverses:** - $g^{-1}(x) = \frac{2x - 18}{7}$ - $f^{-1}(x) = x - 3$ - $f^{-1}(x) = -x + 3$ - $f^{-1}(x) = \frac{x}{4}$ - $f^{-1}(x) = -5x - 5$ - $g^{-1}(x) = \frac{x + 1}{x}$ - $f^{-1}(x) = \sqrt[3]{-\frac{x - 1}{2}}$ - $g^{-1}(x) = -3x - 5$ --- **Desmos functions for graphing:** - For each function $f(x)$ and its inverse $f^{-1}(x)$, plot both on the same coordinate grid.