Question

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)`?

Answer 1

`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`