Using the graph of f(x) and g(x), where g(x) = f(k⋅x), determine the value of k.
a.) 3
b.) 1/3
c.) -1/3
d.) -3

Option B is correct. Using the graph of f(x) and g(x), where g(x) = f(k⋅x), the value of k is 1/3

First, we need to get the equation of each line generally expressed as

y = mx + b

For the blue line, using the coordinate points (1, 2) and (1.5, 5)

Get the slope

m = 5-2/1.5-1

m = 3/0.5

m = 6

Get the y-intercept

2 = 6(1) + b

2 - 6 = b

b = -4

The equation will be g(x) = 6x - 4

For the red line, using the coordinate points (3, 2) and (2, 0)

Get the slope

m = 0-2/2-3

m = -2/-1

m = 2

Get the y-intercept

2 = 2(3) + b

2 - 6 = b

b = -4

The equation will be f(x) = 2x - 4

f(x*k) = 2(x*k)- 4

Equating g(x) to f(k⋅x)

6x + 4 = 2(x*k)- 4

On comparing;

6x = 2xk

6 = 2k

k = 1/3

Hence using the graph of f(x) and g(x), where g(x) = f(k⋅x), the value of k is 1/3

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  A.)  3

Step-by-step explanation:

In the equation g(x) = f(k·x), the factor k is a horizontal compression factor. Here the graph of g is the graph of f compressed by a factor of 3.

The point (3, 2) on the graph of f(x) becomes the point (1, 2) on the graph of g(x). The point (1, 2) is a factor of 3 closer to the y-axis than the point (3, 2).

It may be easier to think of k as the reciprocal of the horizontal expansion (dilation) factor. The function g is horizontally dilated by a factor of 1/3 from function f, so k = 1/(1/3) = 3.