Question

Given \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}-1,\) evaluate each expression. (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(g(f(0))\) (d) \(f(g(-4))\) (e) \(f(g(x))\) (f) \(g(f(x))\)

Short answer

(a) The result of \(f(g(1))\) is \(0\). \n (b) The result of \(g(f(1))\) is also \(0\).\n (c) The result of \(g(f(0))\) is \(-1\).\n (d) \(f(g(-4))\) equals to \(\sqrt{15}\).\n (e) \(f(g(x))\) equals to \(\sqrt{x^{2}-1}\).\n (f) \(g(f(x))\) equals to \(x-1\).

Step-by-step solution

Evaluate part (a): \(f(g(1))\)

First insert \(1\) into \(g(x)\) to get \(g(1) = (1)^{2}-1 = 0\). Then, insert this result into \(f(x)\) to get \(f(g(1)) = f(0) = \sqrt{0} = 0.\)

Evaluate part (b): \(g(f(1))\)

First insert \(1\) into \(f(x)\) to get \(f(1) = \sqrt{1} = 1\). Then, insert this result into \(g(x)\) to get \(g(f(1)) = g(1) = (1)^{2}-1 = 0.\)

Evaluate part (c): \(g(f(0))\)

First insert \(0\) into \(f(x)\) to get \(f(0) = \sqrt{0} = 0\). Then, insert this result into \(g(x)\) to get \(g(f(0)) = g(0) = (0)^{2}-1 = -1.\)

Evaluate part (d): \(f(g(-4))\)

First, insert \(-4\) into \(g(x)\) to get \(g(-4) = (-4)^{2}-1 = 15\). Then, insert this result into \(f(x)\) to get \(f(g(-4)) = f(15) = \sqrt{15}\).

Evaluate part (e): \(f(g(x))\)

In this part, we insert \(g(x)\) into \(f(x)\). This results in \(f(g(x)) = f(x^{2}-1) = \sqrt{x^{2}-1}\).

Evaluate part (f): \(g(f(x))\)

In this part, we insert \(f(x)\) into \(g(x)\). This results in \(g(f(x)) = g(\sqrt{x}) = (\sqrt{x})^{2}-1 = x-1.\)