For problems 1-4, let \(f(x)=12x^{\frac{4}{5}}\), \(g(x)=3x^{\frac{1}{4}}\), and \(h(x)=-7x^{\frac{4}{5}}\) to find the following:
1) \((f+h)(x)\)
2) \((h-f)(x)\)
3) \((g \cdot h)(x)\)
4) \(\left(\frac{f}{g}\right)(x)\)
For problems 5-9, let \(f(x)=x^2+6x+8\) and \(g(x)=x+4\) to find the following:
5) \((f \cdot g)(x)\)
6) \(\left(\frac{f}{g}\right)(x)\)
7) \(g(f(x))\)
8) \(f(g(x))\)
9) Using your answers to questions 7 and 8, explain why compositions of functions are not commutative.
10) Given \(\large\frac{1}{3}x\strut^{\frac{5}{2}}\), determine the function \(g(x)\) such that:
a) \((f+g)(x)=x\strut^{\frac{5}{2}}\)
b) \((f \cdot g)(x)=3x\strut^{\frac{17}{6}}\)
11) Let \(f(x)=9x^5\) and \(g(x)=x\strut^{\frac{1}{2}}\), find:
a) \(f(g(x))\)
b) \(g(f(x))\)
12) If \(f(x)=10x\strut^{\frac{3}{4}}-2\) and \(g(x)=2x\strut^{\frac{3}{4}}-4\), what is \((f-g)(x)\)?
a) \(8x\strut^{\frac{3}{4}}-6\) b) \(-8x\strut^{\frac{3}{4}}-2\) c) \(-8x\strut^{\frac{3}{4}}-6\) d) \(8x\strut^{\frac{3}{4}}+2\)
13) Describe and correct the error in the student's work below.
Problem:
Given \(g(x)=x+5\) and \(h(x) = 3x-1\), find \(h(g(x))\).
Student Work:
\(\begin{align}h(g(x))&=(3x-1)(x+5)\\&=3x^2+15x-x-5\\&=3x^2+14x-5\end{align}\)
14) Let \(f(x)=2x^2-x+4\) and \(g(x)=4x-7\), find:
a) \(f(g(4))\)
b) \(g(f(-2))\)
15) Let \(f(x)=9x\strut^{\frac{1}{2}}\) and \(g(x)=5x\strut^{\frac{1}{3}}\), find \(g(f(9))\).
Review
For Problems 16-17, simplify each expression in the real numbers.
16) \(\sqrt[3]{56x^{12}y^{20}z^4}\)
17) \(5\sqrt[4]{48}+2\sqrt[4]{243}\)
18) Factor the following expressions completely over the set of real numbers:
a) \(8x^3-1\)
b) \(4x^2+25\)
c) \(9x^4+24x^2+16\)
Solution Bank
1) \((f+h)(x)\)
2) \((h-f)(x)\)
3) \((g \cdot h)(x)\)
4) \(\left(\frac{f}{g}\right)(x)\)
For problems 5-9, let \(f(x)=x^2+6x+8\) and \(g(x)=x+4\) to find the following:
5) \((f \cdot g)(x)\)
6) \(\left(\frac{f}{g}\right)(x)\)
7) \(g(f(x))\)
8) \(f(g(x))\)
9) Using your answers to questions 7 and 8, explain why compositions of functions are not commutative.
10) Given \(\large\frac{1}{3}x\strut^{\frac{5}{2}}\), determine the function \(g(x)\) such that:
a) \((f+g)(x)=x\strut^{\frac{5}{2}}\)
b) \((f \cdot g)(x)=3x\strut^{\frac{17}{6}}\)
11) Let \(f(x)=9x^5\) and \(g(x)=x\strut^{\frac{1}{2}}\), find:
a) \(f(g(x))\)
b) \(g(f(x))\)
12) If \(f(x)=10x\strut^{\frac{3}{4}}-2\) and \(g(x)=2x\strut^{\frac{3}{4}}-4\), what is \((f-g)(x)\)?
a) \(8x\strut^{\frac{3}{4}}-6\) b) \(-8x\strut^{\frac{3}{4}}-2\) c) \(-8x\strut^{\frac{3}{4}}-6\) d) \(8x\strut^{\frac{3}{4}}+2\)
13) Describe and correct the error in the student's work below.
Problem:
Given \(g(x)=x+5\) and \(h(x) = 3x-1\), find \(h(g(x))\).
Student Work:
\(\begin{align}h(g(x))&=(3x-1)(x+5)\\&=3x^2+15x-x-5\\&=3x^2+14x-5\end{align}\)
14) Let \(f(x)=2x^2-x+4\) and \(g(x)=4x-7\), find:
a) \(f(g(4))\)
b) \(g(f(-2))\)
15) Let \(f(x)=9x\strut^{\frac{1}{2}}\) and \(g(x)=5x\strut^{\frac{1}{3}}\), find \(g(f(9))\).
Review
For Problems 16-17, simplify each expression in the real numbers.
16) \(\sqrt[3]{56x^{12}y^{20}z^4}\)
17) \(5\sqrt[4]{48}+2\sqrt[4]{243}\)
18) Factor the following expressions completely over the set of real numbers:
a) \(8x^3-1\)
b) \(4x^2+25\)
c) \(9x^4+24x^2+16\)
Solution Bank