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

Question 1: Operations on functions (2 points)

1. \((f+g)(x)=x^{2}-3x+2x+5=\)\(x^{2}-x+5\).
2. \(f(2)=4-6=-2\), \(g(2)=9\), so \((fg)(2)=\)-18.

Scoring: 1 point for each part; in part (b) award the point only for −18 with correct values of f(2) and g(2).

Question 2: Evaluate compositions (2 points)

1. \(g(3)=7\), so \(f(7)=22\): 22.
2. \(f(-1)=-2\), so \(g(-2)=4-2=\)2.

Scoring: 1 point for each part; each point needs the correct inner value and the correct final value.

Question 3: Composite and its domain (2 points)

1. \(f(g(x))=\frac{2}{x-3}\): \(\frac{2}{x-3}\).
2. The denominator cannot be 0, so \(x\neq 3\).

Scoring: 1 point for the composite, 1 point for excluding x = 3.

Question 4: Find an inverse (2 points)

Write \(y=\frac{5x-1}{2}\), swap: \(x=\frac{5y-1}{2}\). Then \(2x=5y-1\), \(5y=2x+1\), so \(f^{-1}(x)=\frac{2x+1}{5}\). Check: \(f(1)=2\) and \(f^{-1}(2)=\frac{5}{5}=1\).

Scoring: 1 point for swapping the variables, 1 point for the correct solved form.

Question 5: Verify inverse functions (2 points)

\(f(g(x))=\frac{3x-12}{3}+4=x-4+4=x\) (1 point).

\(g(f(x))=3(\frac{x}{3}+4)-12=x+12-12=x\) (1 point).

Both compositions equal \(x\), so they are inverses.

Scoring: 1 point for each composition worked correctly.

Question 6: Inverse from a graph (2 points)

1. Swap coordinates: \(A^{\prime}(-3,0)\), \(B^{\prime}(1,2)\), \(C^{\prime}(3,3)\).
2. The line has slope \(\frac{1-(-3)}{2-0}=2\) and \(y\)-intercept \(-3\), so \(f(x)=2x-3\). Swapping gives \(x=2y-3\), so \(f^{-1}(x)=\frac{x+3}{2}\).

Scoring: 1 point for the three swapped points; 1 point for the correct formula (accept an equivalent form).

Question 7: Restrict the domain (3 points)

1. The horizontal line \(y=0\) meets the graph twice, at \(x=-3\) and \(x=1\), so \(f\) is not one-to-one.
2. Restrict to \(x\ge -1\). Write \(x=(y+1)^{2}-4\), so \(y+1=\sqrt{x+4}\) and \(f^{-1}(x)=-1+\sqrt{x+4}\).
3. Domain: \(x\ge -4\). Range: \(y\ge -1\).

Scoring: 1 point for the two-point horizontal line argument; 1 point for a valid restriction with the correct inverse; 1 point for domain and range (accept the restriction x ≤ −1 only with the matching inverse).

Question 8: Taxi fare inverse (3 points)

1. Write \(F=2.50+1.75m\), so \(1.75m=F-2.50\) and \(m(F)=\frac{F-2.50}{1.75}\).
2. \(m(20)=\frac{17.50}{1.75}=10\): 10 miles. Check: \(2.50+1.75(10)=20\).
3. The inverse takes a fare in dollars and returns the length of the ride in miles.

Scoring: 1 point for each part.

Question 9: Growing circle (2 points)

1. \(A(r(t))=\pi(4t)^{2}=\)\(16\pi t^{2}\).
2. \(A(r(3))=16\pi(9)=\)\(144\pi\) square inches. Check: \(r(3)=12\) and \(\pi\cdot 144=144\pi\).

Scoring: 1 point for each part.

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