Question

Please help asap! 30 points

Given that f(x)=11, g(x)=x^2-6x+3, and h(x)= -x+4, find the function (g •h)(x).

Answer 1

`(g \cdot h)(x)=x^2-2x+23`

Explanation:

For composite functions, it's important to understand what the functions mean:

`(g\cdot h)(x)` which is read as "g of h, of x" means
`g ( \text{ }h(x) \text{ })` which is read as "g of, h of x" (with slight pauses at the comma). This means that x goes into the h function, and the output of the h function goes into the g function.

Putting "x" into the h function

`h(x)=-x+4`

Since it is just "x" going into the h function, the function as written is the output when x is the input.

Putting the h function output, into the g function

`g(x)=x^2-6x+3`

`g(h(x))=(h(x))^2-6(h(x))+3`

Substitute

`g(h(x))=(-x+4)^2+-6(-x+4)+3`

Squaring means the something multiplied by itself

`g(h(x))=(-x+4)*(-x+4)+-6(-x+4)+3`

Use distributive property; (some people know binomial distribution as "FOIL" -- First, Outer, Inner, Last):

`g(h(x))=[(-x)(-x)+4(-x)+4(-x)+4*4)]+[6x+4]+3`

Simplify the binomial terms:

`g(h(x))=[x^2-8x+16]+[6x+4]+3`

Group like terms:

`g(h(x))=x^2-2x+23`

Remember that
`(g\cdot h)(x)` means
`g ( \text{ }h(x) \text{ })`

`(g \cdot h)(x)=x^2-2x+23`

So,
`(g \cdot h)(x)=x^2-2x+23`