The functions g(x) and h(x) are:
[tex]\begin{gathered} g(x)=x^2+5 \\ h(x)=x^2+4 \end{gathered}[/tex]
To find the composition (g°h)(x) we use the definition for the composition of functions:
[tex](g\circ h)(x)=g(h(x))[/tex]
This means that to find the composition, we need to plug in h(x) as the value of x in g(x):
[tex](g\circ h)(x)=(x^2+4)^2+5[/tex]
The next step is to simplify the expression. We use the formula for the binomial squared:
[tex](a+b)^2=a^2+2ab+b^2[/tex]
Applying this rule to the first term of the composite function:
[tex](g\circ h)(x)=(x^2)^2+2(x^2)(4)+4^2+5[/tex]
Simplifying the expression further:
[tex](g\circ h)(x)=x^4+8x^2+16+5[/tex]
Finally, we combine 16+5 and we get 21 at the end of the expression:
[tex](g\circ h)(x)=x^4+8x^2+21[/tex]
The composition of the functions is:
[tex](g\circ h)(x)=x^4+8x^2+21[/tex]
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:
[tex]Domain\text{ of }(g\circ h)\colon(-\infty,\infty)[/tex]
Answer:
Composition of the functions:
[tex](g\circ h)(x)=x^4+8x^2+21[/tex]
Domain:
[tex](-\infty,\infty)[/tex]
