Determine whether the following statement is true or false. Explain. The graph of \( f(x)=-x^{5} \) is a reflection in the \( x \)-axis of the graph of \( g(x)=x^{5} \). a \( \quad \) False, because the graph of \( y=f(-x) \) is a reflection in the \( y \)-axis of the graph of \( y \) c False, because the graph of \( y=f(-x) \) is a reflection in the \( y \)-axis of the graph of \( y= \) d \( y=-f(x) \) is a reflection in the \( y \)-axis of the graph of \( y= \) a because the graph of \( y=-f(x) \) is a reflection in the \( x \)-axis of the graph of \( y= \)

The statement is **true**. To understand why, let's analyze the functions involved: 1. The function \( g(x) = x^5 \) is a standard polynomial function that is odd and symmetric about the origin. Its graph passes through the origin and rises steeply in the first quadrant while falling steeply in the third quadrant. 2. The function \( f(x) = -x^5 \) is simply the negative of \( g(x) \). This means that for every point \( (x, g(x)) \) on the graph of \( g(x) \), there is a corresponding point \( (x, f(x)) = (x, -g(x)) \) on the graph of \( f(x) \). This relationship indicates that the graph of \( f(x) \) is indeed a reflection of the graph of \( g(x) \) across the \( x \)-axis. To clarify the options provided: - The statement "the graph of \( f(x) = -x^5 \) is a reflection in the \( x \)-axis of the graph of \( g(x) = x^5 \)" is true because reflecting a point \( (x, y) \) across the \( x \)-axis results in the point \( (x, -y) \). The other options provided seem to be incomplete or incorrectly phrased, but they do not affect the truth of the original statement. Thus, the correct conclusion is that the statement is **true**.