Question

Given the definitions of f(x) and g(x) below, find the value of f(g(-4)). f(x) = x2 - 7x + 11 g(x) = 2x + 8

Answer 1

We can solve this problem using two methods, we can find the new function f(g(x)), or we can find the value of g(-4) and then use this value at f(x). I will do the easiest one first, and after we can find the function f(g(x)).

1st Method - find g(-4)

We can easily find g(-4) doing

`\begin{gathered} g(x)=2x+8 \\ g(-4)=2\cdot(-4)+8 \\ g(-4)=-8+8 \\ g(-4)=0 \end{gathered}`

We put -4 where we had x and do all the calculus, now we know that g(-4) = 0 we can put it in f(x)

`\begin{gathered} f(x)=x^2^{}-7x+11 \\ f(g(-4))=f(0)=0^2-7\cdot0+11 \\ f(g(-4))=11 \end{gathered}`

Therefore f(g(-4)) = 11. That's the easiest way to solve the exercise, now I'll show another way to solve, which requires more algebra but solves more problems

2nd Method - find the function f(g(x))

When we do f(g(x)) it means that where we have "x" in the f function we are going to replace by g(x), in other words, 2x+8, then

`\begin{gathered} f(x)=x^2-7x+11 \\ g(x)=2x+8 \\ f(g(x))=(2x+8)^2-7(2x+8)+11 \end{gathered}`

See that we have a lot of simplification to do, that's the bad part of this method, doing that we have

`\begin{gathered} f(g(x))=4x^2+32x+64-14x-56+11 \\ f(g(x))=4x^2+18x+19 \\ \end{gathered}`

Now using this equation we can put x = -4 and we will find again that the answer is 11

`\begin{gathered} f(g(-4))=4(-4)^2+18(-4)+19 \\ f(g(-4))=11 \end{gathered}`

Now we can confirm that our answer is 11