The answers in the box are:
- First box: [tex]\mathbf{x^2-8x+17}[/tex]
- Second box: [tex]\mathbf{(-\infty,\infty)}[/tex].
Given that the functions are
[tex]g(x)=x^2+1[/tex]
[tex]h(x)=x-4[/tex]
Since clearly g(x) and h(x) exist for all real values of x so the domain of g(x) and h(x) both are all real numbers.
Now the composition of function is given by,
[tex](g\circ h)(x)=g(h(x))[/tex]
[tex]=g(x-4)[/tex]
[tex]=(x-4)^2+1[/tex]
[tex]=x^2-8x+16+1[/tex], since we know that [tex](a-b)^2=a^2-2ab+b^2[/tex]
[tex]=x^2-8x+17[/tex]
So, [tex](g\circ h)(x)=x^2-8x+17[/tex]
It is a polynomial function so it is defined for all real values of x.
Hence the domain of composition is all real numbers, that is [tex](-\infty,\infty)[/tex]
Learn more about Composition of functions here -
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