Question

shifting the cube root function quick check
what is the effect on the x-intercept of the graph of $f(x) = \sqrt3{x}$ when $f(x)$ is replaced by $f(x + 5)$?
\bigcirc the x-intercept shifts to the left by a distance of 5.
\bigcirc the x-intercept shifts to the right by a distance of 5.
\bigcirc the x-intercept shifts up by a distance of 5.
\bigcirc the x-intercept shifts down by a distance of 5.

Explanation:

Step1: Find x-intercept of \( f(x)=\sqrt[3]{x} \)

Set \( f(x) = 0 \), so \( \sqrt[3]{x}=0 \). Cubing both sides, \( x = 0 \). The x-intercept is \( (0,0) \).

Step2: Find x-intercept of \( f(x + 5)=\sqrt[3]{x + 5} \)

Set \( f(x + 5)=0 \), so \( \sqrt[3]{x + 5}=0 \). Cubing both sides, \( x + 5 = 0 \), so \( x=-5 \). The x-intercept is \( (-5,0) \).

Step3: Analyze the shift

From \( x = 0 \) (original) to \( x=-5 \) (new), the x-intercept moves left by \( 0 - (-5)=5 \) units.

Answer:

The x-intercept shifts to the left by a distance of 5. (Corresponding option: The x-intercept shifts to the left by a distance of 5.)