The value of k, for which the graph of g(x) = f(x) + k holds, where
f(x) = -2x-2, is k = 1.
What are linear functions?
Any function of form f(x) = mx + b, is a linear function, where m and b are the slopes and the y-intercept of the line they represent.
How do we solve the given question?
We are given two functions, f(x) = -2x - 2, and g(x) = f(x) + k.
The graph of g(x) is given, and we are asked to determine the value of k.
Since g(x) is a linear function (which can be seen in the graph), it will be in the form g(x) = mx + b.
Now, we will determine m, using the formula m = {(y2-y1)/(x2-x1)} for a line passing through the points (x1, y1), and (x2, y2).
We take the two points from the graph: (0, -1), and (-1, 1).
∴ Slope m = (1 - (-1))/(-1 - 0) = 2/(-1) = -2.
Given g(x) = f(x) + k,
or, g(x) = -2x - 2 + k. Since, the slope m = -2 (calculated), k is not a term in x.
∴ (k - 2) is the y-intercept of g(x). From the graph, the y-intercept is -1.
∴ k - 2 = -1
or, k = -1 + 2 = 1.
∴ The value of k, for which the graph of g(x) = f(x) + k holds, where
f(x) = -2x-2, is k = 1.
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