4.1 - Verifying Invers... 4.1 - Verifying Inverses (Linear and Cubic) Use the theorem on inverse functions to pro and sketch the graphs of f and g on the same 1. \( f(x)=\frac{1}{2} x-3 ; g(x)=2 x+6 \) 2. \( f(x)=\sqrt[3]{x}-1 ; g(x)=(x+1)^{3} \) 3. \( f(x)=\frac{x+2}{3} ; g(x)=3 x-2 \) 4. \( f(x)=\sqrt[3]{x+2} ; g(x)=x^{3}-2 \)

Sure! Let's verify whether each pair of functions \( f \) and \( g \) are inverses of each other using the theorem on inverse functions. We'll also discuss how their graphs relate graphically. ### **Theorem on Inverse Functions** Two functions \( f \) and \( g \) are inverses of each other if and only if: 1. \( f(g(x)) = x \) for all \( x \) in the domain of \( g \). 2. \( g(f(x)) = x \) for all \( x \) in the domain of \( f \). When two functions are inverses, their graphs are **reflections** of each other across the line \( y = x \). Let's apply this to each pair of functions provided. --- ## **1. \( f(x) = \frac{1}{2}x - 3 \) and \( g(x) = 2x + 6 \)** ### **Verification:** - **Calculate \( f(g(x)) \):** \[ f(g(x)) = f(2x + 6) = \frac{1}{2}(2x + 6) - 3 = x + 3 - 3 = x \] - **Calculate \( g(f(x)) \):** \[ g\left(\frac{1}{2}x - 3\right) = 2\left(\frac{1}{2}x - 3\right) + 6 = x - 6 + 6 = x \] Since both compositions yield \( x \), \( g \) is indeed the inverse of \( f \). ### **Graphical Interpretation:** - The graph of \( f \) is a straight line with slope \( \frac{1}{2} \) and y-intercept at \( -3 \). - The graph of \( g \) is a straight line with slope \( 2 \) and y-intercept at \( 6 \). - These lines are reflections of each other across the line \( y = x \). --- ## **2. \( f(x) = \sqrt[3]{x} - 1 \) and \( g(x) = (x + 1)^3 \)** ### **Verification:** - **Calculate \( f(g(x)) \):** \[ f\big(g(x)\big) = f\left((x + 1)^3\right) = \sqrt[3]{(x + 1)^3} - 1 = x + 1 - 1 = x \] - **Calculate \( g(f(x)) \):** \[ g\big(f(x)\big) = g\left(\sqrt[3]{x} - 1\right) = \left(\sqrt[3]{x} - 1 + 1\right)^3 = \left(\sqrt[3]{x}\right)^3 = x \] Both compositions yield \( x \), so \( g \) is the inverse of \( f \). ### **Graphical Interpretation:** - The graph of \( f \) is the cube root function shifted down by 1 unit. - The graph of \( g \) is the cubic function shifted left by 1 unit. - These graphs are reflections of each other across the line \( y = x \). --- ## **3. \( f(x) = \frac{x + 2}{3} \) and \( g(x) = 3x - 2 \)** ### **Verification:** - **Calculate \( f(g(x)) \):** \[ f(g(x)) = f(3x - 2) = \frac{(3x - 2) + 2}{3} = \frac{3x}{3} = x \] - **Calculate \( g(f(x)) \):** \[ g\left(\frac{x + 2}{3}\right) = 3\left(\frac{x + 2}{3}\right) - 2 = (x + 2) - 2 = x \] Since both compositions equal \( x \), \( g \) is the inverse of \( f \). ### **Graphical Interpretation:** - The graph of \( f \) is a straight line with slope \( \frac{1}{3} \) and y-intercept at \( \frac{2}{3} \). - The graph of \( g \) is a straight line with slope \( 3 \) and y-intercept at \( -2 \). - These lines are reflections of each other across the line \( y = x \). --- ## **4. \( f(x) = \sqrt[3]{x + 2} \) and \( g(x) = x^3 - 2 \)** ### **Verification:** - **Calculate \( f(g(x)) \):** \[ f\big(g(x)\big) = f(x^3 - 2) = \sqrt[3]{(x^3 - 2) + 2} = \sqrt[3]{x^3} = x \] - **Calculate \( g(f(x)) \):** \[ g\big(f(x)\big) = g\left(\sqrt[3]{x + 2}\right) = \left(\sqrt[3]{x + 2}\right)^3 - 2 = (x + 2) - 2 = x \] Both compositions result in \( x \), confirming that \( g \) is the inverse of \( f \). ### **Graphical Interpretation:** - The graph of \( f \) is the cube root function shifted left by 2 units. - The graph of \( g \) is the cubic function shifted down by 2 units. - These graphs are reflections of each other across the line \( y = x \). --- ## **Summary of Graphs** For each pair of functions \( f \) and \( g \): - **Verification:** We showed \( f(g(x)) = x \) and \( g(f(x)) = x \), proving that \( g \) is the inverse of \( f \). - **Graphical Relationship:** The graphs of each pair \( f \) and \( g \) are mirror images across the line \( y = x \), illustrating their inverse relationship. If you'd like to see the graphs plotted, you can use graphing software or tools like Desmos to visualize these functions and their inverses.