If f(x)=1/2x and g(x) is a linear function, such that x=−3 and x=−1 when f(x)=g(x). Write a possible function for g(x)?

1 Answer
Jun 2, 2018

g(x) = f(x) = 1/2x

This is the only answer for g(x)

Explanation:

Intuition

Since f(x)=g(x) at two points x=-3,-1, we can tell that g(x) passes through the two points (-3,-3/2), (-1,-1/2)

But if g(x) is linear, that means that its graph is a straight line.

And since there is only one way to draw a straight line through two points, there is only one answer for g(x), and since f(x) is a straight line passing through those two points as well, g(x) has to be equal to f(x)

graph{(y-1/2x)=0 [-5, 2, -5, 2]}

More Rigorous Proof

Since g(x) is linear, we let g(x)=ax+b for constants a & b, then form equations of a & b using the given conditions

Substituting f(-1)=g(-1), we have

1/2(-1)=a(-1)+b

-1/2=-a+b

Substituting f(-3)=g(-3), we get

1/2(-3)=a(-3)+b

-3/2=-3a+b

If you solve these two simultaneous equations, you get a=1/2,b=0 as the only solution

Hence g(x) can only be 1/2x