Function Operations, Composition and Inverses: Algebra 2 Math Worksheet Answer Key – PDF Download

Problem 1: Add and subtract functions

1. \((f+g)(x)=3x-4+x^{2}+2x=\)\(x^{2}+5x-4\).
2. \((f-g)(x)=3x-4-x^{2}-2x=\)\(-x^{2}+x-4\). Distribute the minus sign to both terms of \(g\).

Problem 2: Multiply functions

1. \((fg)(x)=(x+3)(2x-5)=2x^{2}-5x+6x-15=\)\(2x^{2}+x-15\).
2. \((fg)(4)=2(16)+4-15=21\). Check: \(f(4)\cdot g(4)=7\cdot 3=21\). 21

Problem 3: Divide functions and state the domain

\(\frac{x^{2}-9}{x-3}=\frac{(x-3)(x+3)}{x-3}=x+3\). Because \(g(3)=0\), \(x=3\) is not allowed. \((\frac{f}{g})(x)=x+3,\ x\neq 3\); the domain is all real numbers except 3. Check: \(\frac{f(5)}{g(5)}=\frac{16}{2}=8=5+3\).

Problem 4: Operations from graphs

Read the values: \(f(1)=2\), \(g(1)=5\), \(f(5)=6\), \(g(5)=1\), \(f(3)=4\), \(g(3)=3\).

1. \((f+g)(1)=2+5=\)7
2. \((f-g)(5)=6-1=\)5
3. \((fg)(3)=4\cdot 3=\)12

Notice that \(f(x)+g(x)=(x+1)+(6-x)=7\) for every \(x\), so \((f+g)(x)\) is the horizontal line \(y=7\).

Problem 5: Evaluate a composition

1. \(g(3)=10\), so \(f(g(3))=f(10)=2(10)-5=\)15.
2. \(f(3)=1\), so \(g(f(3))=g(1)=1+1=\)2. The order changes the answer.

Problem 6: Write composite functions

1. \(f(g(x))=3x^{2}+4\): \(3x^{2}+4\).
2. \(g(f(x))=3(x+4)^{2}=3(x^{2}+8x+16)=\)\(3x^{2}+24x+48\).

Problem 7: Trace a value through a machine

1. \(g(2)=3(2)-1=5\) and \(f(5)=25\). \(f(g(2))=25\).
2. \(f(g(x))=(3x-1)^{2}=\)\(9x^{2}-6x+1\). At \(x=2\): \(36-12+1=25\), which matches.

Problem 8: Domain of a composition

1. \(f(g(x))=\sqrt{x-5}\) needs \(x-5\ge 0\), so the domain is \(x\ge 5\).
2. \(g(f(x))=\sqrt{x}-5\) needs \(x\ge 0\), so the domain is \(x\ge 0\).

Problem 9: Decompose a function

1. Inner \(g(x)=5x-2\), outer \(f(x)=x^{3}\). Check: \(f(g(x))=(5x-2)^{3}\).
2. Inner \(g(x)=x^{2}+1\), outer \(f(x)=\frac{1}{x}\). Check: \(f(g(x))=\frac{1}{x^{2}+1}\). Other correct choices exist.

Problem 10: Inverse of a linear function

Write \(y=4x-9\), swap: \(x=4y-9\), so \(4y=x+9\) and \(f^{-1}(x)=\frac{x+9}{4}\). Check: \(f(5)=11\) and \(f^{-1}(11)=\frac{20}{4}=5\).

Problem 11: Inverse with fractions

Write \(y=-\frac{2}{3}x+4\) and swap: \(x=-\frac{2}{3}y+4\). Then \(x-4=-\frac{2}{3}y\), so \(y=-\frac{3}{2}(x-4)=-\frac{3}{2}x+6\). \(f^{-1}(x)=-\frac{3}{2}x+6\). Check: \(f(3)=2\) and \(f^{-1}(2)=-3+6=3\).

Problem 12: Verify inverses by composition

\(f(g(x))=5\cdot\frac{x-3}{5}+3=x-3+3=x\) and \(g(f(x))=\frac{(5x+3)-3}{5}=x\). Both compositions give \(x\), so they are inverses.

Problem 13: Are they inverses?

\(f(g(x))=2(\frac{x}{2}-6)+6=x-12+6=x-6\), which is not \(x\). They are not inverses. The inverse of \(f\): \(x=2y+6\), so \(f^{-1}(x)=\frac{x-6}{2}=\frac{x}{2}-3\).

Problem 14: Inverse with a restricted domain

Write \(x=y^{2}+3\), so \(y^{2}=x-3\). Since \(y\ge 0\), \(f^{-1}(x)=\sqrt{x-3}\), with domain \(x\ge 3\) (the range of \(f\)). \(f^{-1}(12)=\sqrt{9}=\)3. Check: \(f(3)=12\).

Problem 15: Inverse of a square root function

Write \(x=\sqrt{y+2}\). Square both sides: \(x^{2}=y+2\), so \(f^{-1}(x)=x^{2}-2\). The outputs of \(f\) are \(\ge 0\), so the domain is \(x\ge 0\). Check: \(f(7)=\sqrt{9}=3\) and \(f^{-1}(3)=9-2=7\).

Problem 16: Reflect points over y = x

1. Swap each pair: \((1,-2)\), \((2,0)\), \((3,2)\), \((4,4)\).
2. The graph of \(f\) has slope \(\frac{1}{2}\) and \(y\)-intercept 2, so \(f(x)=\frac{1}{2}x+2\). Swap and solve: \(x=\frac{1}{2}y+2\), so \(f^{-1}(x)=2x-4\). It passes through all four swapped points.

Problem 17: Horizontal line test

1. Yes. At \((0,3)\) and \((4,3)\), since \((0-2)^{2}-1=3\) and \((4-2)^{2}-1=3\).
2. No. Two different inputs give the output 3, so \(f\) is not one-to-one and its inverse would not be a function.
3. Restrict to \(x\ge 2\) (or \(x\le 2\)). Each horizontal line then meets the graph at most once.

Problem 18: Inverse after restricting the domain

Write \(x=(y-2)^{2}-1\), so \((y-2)^{2}=x+1\). Because \(y\ge 2\), \(y-2=\sqrt{x+1}\) and \(f^{-1}(x)=2+\sqrt{x+1}\), domain \(x\ge -1\). Check: \(f(5)=9-1=8\) and \(f^{-1}(8)=2+\sqrt{9}=5\).

Problem 19: Inverse of a rational function

Write \(x=\frac{2y}{y+3}\). Then \(xy+3x=2y\), so \(3x=2y-xy=y(2-x)\) and \(f^{-1}(x)=\frac{3x}{2-x}\), \(x\neq 2\). Check: \(f(3)=\frac{6}{6}=1\) and \(f^{-1}(1)=\frac{3}{1}=3\).

Problem 20: Compositions that are equal

\(f(g(x))=2x^{2}-1\) and \(g(f(x))=(2x-1)^{2}=4x^{2}-4x+1\). Set equal: \(2x^{2}-1=4x^{2}-4x+1\), so \(0=2x^{2}-4x+2=2(x-1)^{2}\). \(x=1\) is the only solution. Check: \(f(g(1))=f(1)=1\) and \(g(f(1))=g(1)=1\).

Problem 21: Temperature conversion

1. Write \(F=\frac{9}{5}C+32\) and solve for \(C\): \(F-32=\frac{9}{5}C\), so \(C(F)=\frac{5}{9}(F-32)\).
2. \(C(95)=\frac{5}{9}(63)=35\), so \(95^{\circ}\text{F}=35^{\circ}\text{C}\). \(C(-4)=\frac{5}{9}(-36)=-20\), so \(-4^{\circ}\text{F}=-20^{\circ}\text{C}\). Both points lie on the graph.

Problem 22: Car rental inverse

1. Write \(C=0.45m+25\) and solve for \(m\): \(m(C)=\frac{C-25}{0.45}\).
2. The inverse takes a bill and returns the number of miles driven. \(m(70)=\frac{45}{0.45}=100\), so the bill of $70 covers 100 miles. Check: \(0.45(100)+25=70\).

Problem 23: Discount and coupon order

1. \(c(d(x))=0.8x-5\) and \(d(c(x))=0.8(x-5)=0.8x-4\).
2. Discount first: \(c(d(60))=48-5\), so $43. Coupon first: \(d(c(60))=0.8(55)\), so $44.
3. Taking the 20% off first is cheaper. The difference is \((0.8x-4)-(0.8x-5)=1\), so it saves $1 at every price.

Problem 24: Composition in context

1. \(A(r(t))=\pi(3t)^{2}=\)\(9\pi t^{2}\).
2. \(A(r(2))=9\pi(4)=36\pi\), about 113.1 square inches. Check: \(r(2)=6\) and \(\pi\cdot 36=36\pi\).

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