Question

Composition of two functions: AdvancedFor the real-valued functions g(x) = x° +5 and h(x) = x +4, find the composition goh and specify its domain using interval notat

Answer 1

The functions g(x) and h(x) are:

`\begin{gathered} g(x)=x^2+5 \\ h(x)=x^2+4 \end{gathered}`

To find the composition (g°h)(x) we use the definition for the composition of functions:

`(g\circ h)(x)=g(h(x))`

This means that to find the composition, we need to plug in h(x) as the value of x in g(x):

`(g\circ h)(x)=(x^2+4)^2+5`

The next step is to simplify the expression. We use the formula for the binomial squared:

`(a+b)^2=a^2+2ab+b^2`

Applying this rule to the first term of the composite function:

`(g\circ h)(x)=(x^2)^2+2(x^2)(4)+4^2+5`

Simplifying the expression further:

`(g\circ h)(x)=x^4+8x^2+16+5`

Finally, we combine 16+5 and we get 21 at the end of the expression:

`(g\circ h)(x)=x^4+8x^2+21`

The composition of the functions is:

`(g\circ h)(x)=x^4+8x^2+21`

We also need to find the domain. The domain are the values allowed or possible for the x variable, in this case, we can have any value as the x value without problems. Any value is possible for x. Thus, the Domain is:

`Domain\text{ of }(g\circ h)\colon(-\infty,\infty)`

Answer:

Composition of the functions:

`(g\circ h)(x)=x^4+8x^2+21`

Domain:

`(-\infty,\infty)`