Question

given that $g(x) = f(x) + k$, identify the value of $k$ for the functions $f$ and $g$ shown on the graph.
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Explanation:

Step1: Find a point on f(x)

Looking at the graph, a point on \( f(x) \) could be, for example, when \( x = 0 \), \( f(0)=1 \) (by observing the y - value of the line \( f \) at \( x = 0 \)).

Step2: Find the corresponding point on g(x)

For the same \( x = 0 \), the line \( g(x) \) has a y - value of \( g(0)= - 1 \) (by observing the y - value of the line \( g \) at \( x = 0 \)).

Step3: Use the equation \( g(x)=f(x)+k \)

Substitute \( x = 0 \), \( f(0) = 1 \) and \( g(0)=-1 \) into \( g(x)=f(x)+k \). We get \( - 1=1 + k \).

Step4: Solve for k

Subtract 1 from both sides of the equation \( - 1=1 + k \). So, \( k=-1 - 1=-2 \). We can also check with other points. For example, take a point on \( f(x) \) like \( x = 2 \), \( f(2) = 2 \). The corresponding point on \( g(x) \) at \( x = 2 \) is \( g(2)=0 \). Substitute into \( g(x)=f(x)+k \): \( 0 = 2 + k\), so \( k=-2 \).

Answer:

\( k=-2 \)