Question

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

a.

b.

c.

d.

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 inverse

• The parent function is $y=x^3$, shifted up 4 units.
• When $x=0$, $f^{-1}(0) = 0^3 + 4 = 4$ (y-intercept at $(0,4)$).
• When $f^{-1}(x)=0$, $0 = x^3 + 4 \implies x = \sqrt[3]{-4} \approx -1.59$, so x-intercept near $(-1.6, 0)$.
• The function is a cubic, increasing for all real $x$, with the characteristic cubic "S" shape, shifted upward.

Step3: Match to the graph

Compare to options:

• Option a: Shifts left, does not match.
• Option b: Has the cubic shape, shifted up, matches the key points and behavior of $y=x^3+4$.
• Option c: No upward shift, does not match.
• Option d: Starts at $x=4$, does not match the domain of the inverse (all real numbers).

Answer:

b. <graph of increasing cubic curve shifted upward with y-intercept above the origin>