Question

what is the effect on the graph of $f(x) = x^3$ when $f(x)$ is replaced by $f(-x)$? (1 poin

○ a reflection about the $y$-axis

○ a reflection about the line $y = x$

○ the range changing to only negative values of $y$

○ the domain changing to only negative values of $x$

Explanation:

Step1: Recall reflection rules

For a function \( y = f(x) \), replacing \( x \) with \( -x \) gives \( y = f(-x) \), which is a reflection about the \( y \)-axis.

Step2: Analyze \( f(x)=x^3 \) and \( f(-x) \)

Given \( f(x)=x^3 \), then \( f(-x)=(-x)^3=-x^3 \). The graph of \( f(-x) \) is a reflection of \( f(x) \) over the \( y \)-axis.

• Reflection about \( y = x \) is for inverse functions, not \( f(-x) \), so the second option is wrong.
• The range of \( f(x)=x^3 \) is all real numbers, and the range of \( f(-x)=-x^3 \) is also all real numbers, so the third option is wrong.
• The domain of both \( f(x) \) and \( f(-x) \) is all real numbers, so the fourth option is wrong.

Answer:

a reflection about the \( y \)-axis