Question

we expect to see a ___ for the graph of a composition of a function and its inverse function, if the domain of each is all real numbers. line parabola curve circle 5. find the composition of the function. $f(x)=sqrt{x}$ and $g(x)=x^{2}-3$ find $g(f(x))$. $x + 3$ $x - 3$ $sqrt{x^{2}-3}$ $x-sqrt{3}$ 6. find the composition of the function. $f(x)=x^{\frac{1}{2}}$ and $g(x)=x^{4}+5$ find $g(f(x))$. $x^{2}+5$ $x^{2}+sqrt{5}$ $(x + 5)^{2}$ $sqrt{x+5}$

Explanation:

Question 1:

Step1: Recall function - inverse property

If \(y = f(x)\) and \(x = f^{-1}(y)\), then \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x)) = x\) for all \(x\) in the appropriate domain. The graph of \(y = x\) is a line. So, the graph of a composition of a function and its inverse (when the domain of each is all real - numbers) is the line \(y = x\).

Question 5:

Step1: Substitute \(f(x)\) into \(g(x)\)

Given \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}-3\), to find \(g(f(x))\), we substitute \(f(x)\) into \(g(x)\). Replace \(x\) in \(g(x)\) with \(\sqrt{x}\). So \(g(f(x))=(\sqrt{x})^{2}-3\).

Step2: Simplify the expression

Since \((\sqrt{x})^{2}=x\) for \(x\geq0\), then \(g(f(x))=x - 3\).

Question 6:

Step1: Substitute \(f(x)\) into \(g(x)\)

Given \(f(x)=x^{\frac{1}{2}}\) and \(g(x)=x^{4}+5\), to find \(g(f(x))\), we substitute \(f(x)\) into \(g(x)\). Replace \(x\) in \(g(x)\) with \(x^{\frac{1}{2}}\). So \(g(f(x))=(x^{\frac{1}{2}})^{4}+5\).

Step2: Simplify using exponent rules

Using the rule \((a^{m})^{n}=a^{mn}\), \((x^{\frac{1}{2}})^{4}=x^{\frac{1}{2}\times4}=x^{2}\). Then \(g(f(x))=x^{2}+5\).

Answer:

1. line
2. \(x - 3\)
3. \(x^{2}+5\)