Question

Let $f(x) = 2x^2 + 3x - 9,$ $g(x) = 5x + 11,$ and $h(x) = -3x^2 + 1.$ Find $f(x) - g(x) + h(x).$

Let $f(x) = 4x - 7$, $g(x) = (x + 1)^2$, and $s(x) = f(x) + g(x)$. What is $s(3)$?

Let $f(x) = 3x + 2$ and $g(x) = x^2 - 5x - 1.$ Find $f(g(x)).$

Let $f(x) = 3(x - 6)^2 + 1$. What is the range of $f$? Express your answer with interval notation.

Answer 1

QUESTION 1

Given that:

`f(x)=2x^2+3x-9`,

`g(x)=5x+11`,

and

`h(x)=-3x^2+1`

Then;

`f(x)-g(x)+h(x)=2x^2+3x-9-(5x+11)+(-3x^2+1)`

`f(x)-g(x)+h(x)=2x^2+3x-9-5x-11-3x^2+1`

Group similar terms;

`f(x)-g(x)+h(x)=2x^2-3x^2+3x-5x-11-9+1`

Simplify;

`f(x)-g(x)+h(x)=-x^2-2x-19`

QUESTION 2

Given that;

`f(x)=4x-7`.

`g(x)=(x+1)^2`

and

`s(x)=f(x)+g(x)`

Substitute the functions;

`s(x)=4x-7+(x+1)^2`

Substitute x=3

`s(3)=4(3)-7+(3+1)^2`

`s(3)=12-7+(4)^2`

`s(3)=5+16`

`s(3)=21`

QUESTION 3

Given:

`f(x)=3x+2`

`g(x)=x^2-5x-1`

`f(g(x))=f(x^2-5x-1)`

This implies that;

`f(g(x))=3(x^2-5x-1)+2`

Expand the parenthesis;

`f(g(x))=3x^2-15x-3+2`

`f(g(x))=3x^2-15x-1`

QUESTION 4

The given function is;

`f(x)=3(x-6)^2+1`

Let

`y=3(x-6)^2+1`

`\Rightarrow y-1=3(x-6)^2`

`\Rightarrow (y-1)/(3)=(x-6)^2`

`\Rightarrow \sqrt{(y-1)/(3)}=x-6`

`\Rightarrow x=6+\sqrt{(y-1)/(3)}`

The range is:

`(y-1)/(3)\ge0`

`y-1\ge0`

`y\ge1`

The interval notation is;

`[1,+\infty)`