Question

1. structure match $f(x) = \sqrt3{x + 4}$ with the graph of its inverse.

a. $\bigcirc$ graph with x from -8 to 4, y from -4, blue curve

b. $\bigcirc$ graph with x from -2 to 2, y with 2, blue curve

c. $\bigcirc$ graph with x from -2 to 2, y from -2, blue curve

d. $\bigcirc$ graph with x from -4 to 8, y from -4 to 4, blue curve

Explanation:

Step1: Find inverse of $f(x)$

Start with $y = \sqrt[3]{x + 4}$. Swap $x$ and $y$:
$x = \sqrt[3]{y + 4}$
Cube both sides:
$x^3 = y + 4$
Solve for $y$:
$y = x^3 - 4$
So $f^{-1}(x) = x^3 - 4$

Step2: Identify key features of $f^{-1}(x)$

• Y-intercept: When $x=0$, $y = 0^3 - 4 = -4$, so point $(0, -4)$
• X-intercept: When $y=0$, $x^3 - 4 = 0 \implies x = \sqrt[3]{4} \approx 1.59$, so point $(\sqrt[3]{4}, 0)$
• Shape: Cubic function, increasing for all $x$, matches the general cubic curve shape.

Step3: Match to graphs

Graph c passes through $(0, -4)$ and has the increasing cubic curve shape that matches $y = x^3 - 4$.

Answer:

c. <The graph with x-intercept near 1.5, y-intercept at (0, -4), and increasing cubic curve>