Question

Given that `f(x)=x+2` and `g( f(x) ) = x^2 + 4x-2` then write down `g(f(x))`?

Answer 1

`g(x)=x^2-2.`

Explanation:

Let, `f(x)=y=x+2. :. x=y-2`

Now, given that, `g(f(x))=x^2+4x-2.`

Since, `f(x)=y," this means that, "g(y)=x^2+4x-2.`

But, we know that, `x=y-2.`

Hence, substituting `x=y-2,` in, `g(y),` we get,

`g(y)=(y-2)^2+4(y-2)-2,`

`=y^2-4y+4+4y-8-2.`

`:. g(y)=y^2-6.`

This is same as to say that, `g(x)=x^2-2.`

Answer 2

See below.

Explanation:

If

`f(x) = alpha x + beta`

and

`g(f(x)) = a x^2+b x+ c`

then

`g(x) = m x^2+nx+p`

so

`g(f(x)) = m(alpha x+beta)^2+n(alpha x+beta)+p = a x^2+b x+ c`

or comparing coefficients

`{(beta^2 m + beta n + p=c), (2 alpha beta m + alpha n = b), (m alpha^2=a):}`

and solving for `m,n,p`

#((m = a/alpha^2), (n = (alpha b - 2 a beta)/alpha^2), (p = (beta (a beta-alpha b))/alpha^2 + c))#

Answer 3

`g(x) = x^2-6`

Explanation:

Because of the property `f(f^-1(x))=x`, one can find `g(x)` by evaluating `g(f(x))` at `f^-1(x)`

`g(x) = g(f(f^-1(x)))`

This implies that we must find `f^-1(x)`; we begin with `f(x)`:

`f(x)=x+2`

Then substitute `f^-1(x)` for every x in `f(x)`:

`f(f^-1(x))=f^-1(x)+2`

The left side becomes x by definition:

`x=f^-1(x)+2`

Solve for `f^-1(x)`:

`f^-1(x)= x-2`

To verify that we have `f^-1(x)`, we check `f(f^-1(x)) = x` and `f^-1(f(x)) = x`:

`f(f^-1(x)) = (x - 2)+2 = x`

`f^-1(f(x))= (x+2)-2 = x`

Now that we are sure that we have `f^-1(x)` we evaluate `g(f(f^-1(x)))`:

`g(f(x)) = x^2+ 4x - 2`

Evaluate at `f^-1(x)`

`g(f(f^-1(x))) = (f^-1(x))^2+ 4(f^-1(x)) - 2`

The left side becomes `g(x)` by definition and we substitute the equivalent for `f^-1(x)` into the right side terms:

`g(x) = (x-2)^2+ 4(x-2) - 2`

Expand the square:

`g(x) = x^2-4x+4+ 4(x-2) - 2`

Distribute the 4:

`g(x) = x^2-4x+4+ 4x-8 - 2`

`g(x) = x^2-6`

Answer 4

`g(x) = x^2- 6`

Explanation:

The term "write down" in an exam style question, typically suggest very little working (if any) or effort is required to form the solution:

We know that `f(x)=x+2` so we aim to find `g(f(x))` as a function of `(x+2)`

`g( f(x) ) = x^2 + 4x-2`

We can complete the square to get:

`g( f(x) ) = (x+2)^2 -2^2 -2`
`\ \ \ \ \ \ \ \ \ \ \ = (x+2)^2 -4 -2`
`\ \ \ \ \ \ \ \ \ \ \ = (x+2)^2 -6`

We therefore have:

`g( x+2 ) = (x+2)^2 -6`

Hence:

`g(x) = x^2- 6`