Question

test yourself! practice tool
the graph of functions f(x) and g(x) are shown.
which of the following statements are true? select all that apply.
□ g(x) = -f(x + 3)
□ g(x) = -f(x - 3)
□ g(x) = -f(x) - 3
□ g(x) = -f(x) + 3
□ g(x) = -|x| + 3
□ g(x) = |-x + 3|

Explanation:

Step1: Analyze \( f(x) \)

From the graph, \( f(x) = |x| \) (since it's a V - shaped graph with vertex at the origin, passing through (1,1), ( - 1,1) etc.).

Step2: Analyze \( g(x) \)

The graph of \( g(x) \) is a V - shaped graph opening downward with vertex at (0,3). So, \( g(x)=-|x| + 3 \).

Step3: Check each option

• Option 1: \( g(x)=-f(x + 3) \). Since \( f(x)=|x| \), then \( -f(x + 3)=-|x + 3| \). The vertex of \( -|x + 3| \) is at ( - 3,0), which does not match the vertex of \( g(x) \) (0,3). So this option is incorrect.
• Option 2: \( g(x)=-f(x - 3) \). \( -f(x - 3)=-|x - 3| \). The vertex of \( -|x - 3| \) is at (3,0), which does not match the vertex of \( g(x) \) (0,3). So this option is incorrect.
• Option 3: \( g(x)=-f(x)-3 \). \( -f(x)-3=-|x|-3 \). The vertex of \( -|x|-3 \) is at (0, - 3), which does not match the vertex of \( g(x) \) (0,3). So this option is incorrect.
• Option 4: \( g(x)=-f(x)+3 \). Since \( f(x)=|x| \), \( -f(x)+3=-|x| + 3 \), which matches the equation of \( g(x) \) we derived. So this option is correct.
• Option 5: \( g(x)=-|x| + 3 \). This is exactly the equation of \( g(x) \) we found from the graph. So this option is correct.
• Option 6: \( g(x)=|-x + 3|=|x - 3| \). The graph of \( |x - 3| \) is a V - shaped graph opening upward with vertex at (3,0), which does not match the graph of \( g(x) \) (opening downward, vertex at (0,3)). So this option is incorrect.

Answer:

The correct options are \( \boldsymbol{g(x)=-f(x)+3} \) and \( \boldsymbol{g(x)=-|x| + 3} \) (i.e., the fourth and fifth options: \( \boldsymbol{g(x)=-f(x)+3} \), \( \boldsymbol{g(x)=-|x| + 3} \))