Question

functions $f$ (solid) and $g$ (dashed) are graphed.

graph of two functions f (solid) and g (dashed) on a coordinate plane with x-axis from -8 to 8 and y-axis from -8 to 8. f has segments: from (-8,0) to (-2,-3), then to (3,7), then horizontal to (8,7). g has segments: from (-8,0) to (-2,3), then to (3,-7), then horizontal to (8,-7).

what is the equation of $g$ in terms of $f$?

choose 1 answer:

a $g(x) = f(x)$

b $g(x) = -f(x)$

c $g(x) = f(-x)$

d $g(x) = -f(-x)$

Explanation:

Step1: Analyze reflection over x - axis

A reflection of a function \( y = f(x) \) over the x - axis is given by the transformation \( y=-f(x) \). We can check the key points of the functions \( f \) and \( g \). For example, let's look at a point on \( f(x) \) and the corresponding point on \( g(x) \). If we take a point \((x,y)\) on \( f(x) \), the corresponding point on \( g(x) \) seems to have the same \( x \) - coordinate but the \( y \) - coordinate is the negative of that of \( f(x) \).

Step2: Analyze reflection over y - axis (not needed here)

A reflection over the y - axis is \( y = f(-x) \), but our analysis of the y - coordinates (negative of each other for same x) points towards reflection over x - axis. Let's test with a point. Suppose on \( f(x) \), when \( x = 0 \), \( f(0)=0 \) (from the graph, the solid line passes through the origin). For \( g(x) \), at \( x = 0 \), \( g(0) = 0 \) as well. Now take another point, say when \( x = 2 \), \( f(2) = 7 \) (approx, from the solid line which goes up to \( y = 7\) or so at \( x = 2\)). For \( g(x) \) at \( x = 2 \), the dashed line has a \( y \) - value of \( - 7\) (approx). So \( g(2)=-f(2) \). This suggests that \( g(x)=-f(x) \).

Answer:

B. \( g(x)=-f(x) \)