Question

4 mark for review the piecewise-linear function f, defined on -4 ≤ x ≤ 4, is shown in the graph. the function g is given by g(x) = x + 3. which of the following is the graph of y = f(g(x)) ?

Explanation:

Step1: Analyze the transformation

The function \( g(x)=x + 3 \) represents a horizontal shift. For the composition \( y = f(g(x))=f(x + 3) \), this is a horizontal shift of the graph of \( f(x) \) to the left by 3 units.

Step2: Identify key points of \( f(x) \)

From the graph of \( f(x) \), let's identify some key points. For example, the leftmost horizontal segment starts at \( x=-4 \), the "corner" at \( x = - 2 \), the horizontal segment at \( y=-1 \) from \( x=-1 \) to \( x = 1 \), and the increasing segment starting at \( x = 1 \).

Step3: Apply the horizontal shift

• The point \( x=-4 \) on \( f(x) \) will shift to \( x=-4-3=-7 \) on \( f(x + 3) \).
• The corner at \( x=-2 \) on \( f(x) \) will shift to \( x=-2 - 3=-5 \) on \( f(x + 3) \).
• The start of the increasing segment at \( x = 1 \) on \( f(x) \) will shift to \( x=1-3=-2 \) on \( f(x + 3) \).
• The point \( x = 2 \) (where \( f(x)=0 \)) on \( f(x) \) will shift to \( x=2-3=-1 \) on \( f(x + 3) \).
• The point \( x = 4 \) (where \( f(x)=2 \)) on \( f(x) \) will shift to \( x=4-3 = 1 \) on \( f(x + 3) \).

Now, looking at the option A (the only option partially shown, but based on the shift), the key points should match the shifted positions. The left - most horizontal part should start around \( x=-7 \), the corner around \( x=-5 \), the horizontal part around \( x=-5\) to \( x=-1\) (since \( - 1=-2-3 + 0\)? Wait, no, re - checking: The original horizontal segment of \( f(x) \) from \( x=-4 \) to \( x=-2 \) (y=-2) shifts to \( x=-7 \) to \( x=-5 \). The segment from \( x=-2 \) to \( x=-1 \) (increasing to \( y=-1 \)) shifts to \( x=-5 \) to \( x=-4 \). The horizontal segment from \( x=-1 \) to \( x = 1 \) (y=-1) shifts to \( x=-4 \) to \( x=-2 \). The increasing segment from \( x = 1 \) to \( x = 4 \) shifts to \( x=-2 \) to \( x = 1 \). The graph in option A (the one with left - most point around \( x=-7 \), corner at \( x=-5 \), etc.) matches the shifted graph of \( f(x) \) by \( g(x)=x + 3 \).

Answer:

A (the option with the graph that has the left - most horizontal part starting around \( x = - 7 \), a corner around \( x=-5 \), and the increasing part starting around \( x=-2 \) and going up to \( x = 1 \) with \( y = 2 \) at \( x = 1 \))