Question

Suppose that the functions q and r are defined as follows.

q (x) = x^2+6
r(x) = square x+9

Answer 1

`(r\circ q)(7)=8,\ (r\circ q)(7)=-8`

`(q\circ r)(7)=22`

Explanation:

Composite Function

Given two functions q(x) and r(x), the composite function
`r\circ q(x)` is defined as

`(r\circ q)(x)=r(q(x))`

Similarily

`(q\circ r)(x)=q(r(x))`

The functions are

`q(x)=x^2+6`

`r(x)=√(x+9)`

Compute

`(r\circ q)(x)=√(x^2+6+9)=√(x^2+15)`

Evaluating for x=7

`(r\circ q)(7)=√(7^2+15)=√(64)=\pm 8`

We have two solutions

`\boxed{(r\circ q)(7)=8,\ (r\circ q)(7)=-8}`

Now compute

`(q\circ r)(x)=(√(x+9))^2+6=x+9+6=x+15`

For x=7

`(q\circ r)(7)=7+15=22`

`\boxed{(q\circ r)(7)=22}`