Question

reflect the function $f(x) = x^3$ about the $x$-axis, and translate it 3 units to the left to produce $g(x)$. which equation represents the function $g(x)$?

a $g(x) = -(x + 3)^3$

b $g(x) = -x^3 + 3$

c $g(x) = -(x - 3)^3$

d $g(x) = -x^3 - 3$

Explanation:

Step1: Reflect over x - axis

To reflect a function \( y = f(x) \) over the \( x \) - axis, we use the transformation \( y=-f(x) \). Given \( f(x)=x^{3} \), after reflecting over the \( x \) - axis, the function becomes \( y = -x^{3} \).

Step2: Translate 3 units left

To translate a function \( y = h(x) \) \( a \) units to the left, we use the transformation \( y=h(x + a) \). Here, \( h(x)=-x^{3} \) and \( a = 3 \). So we substitute \( x \) with \( x + 3 \) in \( h(x) \).

Substituting \( x\) with \(x + 3\) in \(y=-x^{3}\), we get \( y=-(x + 3)^{3} \).

Answer:

A. \( g(x)=-(x + 3)^{3} \)