1. **Problem Statement:** Graphically compare the function $f(x) = 3x - 5$ and its inverse. 2. **Step a: Graph the function $f(x) = 3x - 5$.** - This is a linear function with slope 3 and y-intercept -5. 3. **Step b: Find the domain and range of $f(x)$.** - Domain: All real numbers, $(-\infty, \infty)$. - Range: All real numbers, $(-\infty, \infty)$. 4. **Step c: Find the x- and y-intercepts of $f(x)$.** - Y-intercept: Set $x=0$, then $f(0) = 3(0) - 5 = -5$ so $(0, -5)$. - X-intercept: Set $f(x)=0$, solve $0 = 3x - 5 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3}$ so $(\frac{5}{3}, 0)$. 5. **Step d: Find the inverse function $f^{-1}(x)$.** - Start with $y = 3x - 5$. - Swap $x$ and $y$: $x = 3y - 5$. - Solve for $y$: $$x + 5 = 3y \Rightarrow y = \frac{x + 5}{3}$$ - So, $f^{-1}(x) = \frac{x + 5}{3}$. 6. **Step e: Graph the inverse $f^{-1}(x) = \frac{x + 5}{3}$.** - This is also a linear function with slope $\frac{1}{3}$ and y-intercept $\frac{5}{3}$. 7. **Step f: Graph the line $y = x$.** - This line is the reflection line for the function and its inverse. 8. **Step g: Find the domain and range of the inverse function.** - Domain: All real numbers, $(-\infty, \infty)$. - Range: All real numbers, $(-\infty, \infty)$. 9. **Step h: Find the x- and y-intercepts of the inverse function.** - Y-intercept: Set $x=0$, $f^{-1}(0) = \frac{0 + 5}{3} = \frac{5}{3}$ so $(0, \frac{5}{3})$. - X-intercept: Set $f^{-1}(x) = 0$, solve $0 = \frac{x + 5}{3} \Rightarrow x + 5 = 0 \Rightarrow x = -5$ so $(-5, 0)$. **Final summary:** - $f(x) = 3x - 5$ has domain and range $(-\infty, \infty)$, x-intercept $(\frac{5}{3}, 0)$, y-intercept $(0, -5)$. - Its inverse $f^{-1}(x) = \frac{x + 5}{3}$ has domain and range $(-\infty, \infty)$, x-intercept $(-5, 0)$, y-intercept $(0, \frac{5}{3})$. - Both graphs are linear and symmetric about the line $y = x$.