Function Problem

[AlexAmos]

Good Evening all.
I am having trouble solving the following problem

f(x) = x + 5 and h(x) = 2x − 2, find a function g such that g ∘ f = h

I get g(x) = 2x -7 however this is incorrect.

Any help would be appreciated.
Thanks,
Alex.

 

[Dr.Peterson]

First, let's check your answer: g(f(x)) = g(x+5) = 2(x + 5) - 7 = 2x + 3, which is not h(x). So you're right, it's wrong.

How did you find your answer? Perhaps we can correct some step you took?

One way I see to do it is that g = h ∘ f ^-1, which you can see by composing each side with f ^-1.

 

[pka]

Good Evening all.
I am having trouble solving the following problem
f(x) = x + 5 and h(x) = 2x − 2, find a function g such that g ∘ f = h
I get g(x) = 2x -7 however this is incorrect.

Start with \(\displaystyle g(x)=ax+b\) then
\(\displaystyle \begin{align*}g\circ f(x)&=a(x+5)+b \\&=ax+5a+b\\2x-2&=ax+5a+b \end{align*}\)

So \(\displaystyle a=2~\&~b=? \)

 

[Steven G]

Good Evening all.
I am having trouble solving the following problem

f(x) = x + 5 and h(x) = 2x − 2, find a function g such that g ∘ f = h

I get g(x) = 2x -7 however this is incorrect.

Any help would be appreciated.
Thanks,
Alex.

Well what can you do to x+5 to get 2x-2? You can double x+5 to get 2x+10 and then subtract 12 to get 2x-2. So define g(x) to do exactly what we need. That is g(x) = 2x-12.

These type of problems, with practice, can be done in seconds.

I will work out one more for you and then give you one to work on.
.
f(x) = 5x-10 and h(x) = 15x+3
Triple 5x-10 to get 15x-30 and then add 33 to get 15x+3. So g(x) = 3x+33

f(x) = 12x + 2 and h(x) = 6x+2. Find g(x)