Precalculus Combining and Composing Functions Test Solutions and Scoring (PDF)
This function operations test answers for Precalculus covers combining and composing functions (Common Core F-BF.A.1b, F-BF.A.1c), with a free PDF to print.
Precalculus · Common Core State Standards: F-BF.A.1b, F-BF.A.1c
Function operations test answers: worked solutions
Question 1 – Sum and product [2 pts]
\(f(3)=3\), \(g(3)=11\), so \((f+g)(3)=14\). \(f(-1)=3\), \(g(-1)=3\), so \((fg)(-1)=9\).
\(14\) and \(9\)
Scoring: 1 point for each value.
Question 2 – Composite formulas [2 pts]
\(f(g(x))=(3x-2)^{2}+1=9x^{2}-12x+5\). \(g(f(x))=3(x^{2}+1)-2=3x^{2}+1\).
\(9x^{2}-12x+5\) and \(3x^{2}+1\)
Scoring: 1 point for each formula.
Question 3 – Table composition [2 pts]
\(g(2)=2\), \(f(2)=1\). \(g(0)=3\), \(g(3)=1\).
\(f(g(2))=1,\ g(g(0))=1\)
Scoring: 1 point for each value.
Question 4 – Domain of a quotient [2 pts]
\(f\) needs \(x\ge -1\). The quotient excludes \(x=5\).
\([-1,5)\cup(5,\infty)\)
Scoring: 1 point for x >= -1, 1 point for excluding 5.
Question 5 – Domain of a composite [3 pts]
\(g\) needs \(x\ge 0\). Then \(f(g(x))=\frac{1}{\sqrt{x}-2}\), which fails when \(\sqrt{x}=2\), that is \(x=4\).
\([0,4)\cup(4,\infty)\)
Scoring: 1 point for x >= 0, 1 point for finding x = 4, 1 point for the interval.
Question 6 – Decompose [2 pts]
(a) Inside \(x^{2}+4\), outside \(\frac{1}{x}\). (b) Inside \(3x-7\), outside \(|x|\).
(a) \(g=x^{2}+4,\ f=\frac1x\); (b) \(g=3x-7,\ f=|x|\)
Scoring: 1 point for each decomposition.
Question 7 – Evaluate composites [2 pts]
\(f(3)=5\), \(g(5)=25\). \(g(3)=9\), \(f(9)=17\).
\(25\) and \(17\)
Scoring: 1 point for each.
Question 8 – Bakery model [3 pts]
(a) \(R(n(h))=0.75(15h)=11.25h\). (b) \(11.25(8)=90\). (c) \(11.25h=180\), so \(h=16\).
\(11.25h\); \(\$90\); \(16\) hours
Scoring: 1 point for each part.
Question 9 – Inverse check [2 pts]
\(f(g(x))=3\cdot\frac{x+7}{3}-7=x\). \(g(f(x))=\frac{3x-7+7}{3}=x\).
Both equal x
Scoring: 1 point for each composition.
Total score
20 points
Key facts behind this function operations test answers
- (f+g)(x) = f(x) + g(x); the same pattern holds for difference and product.
- (f/g)(x) = f(x)/g(x) requires g(x) not equal to 0.
- (f∘g)(x) = f(g(x)): work from the inside out.
- In general f(g(x)) is not equal to g(f(x)).
- Domain of f(g(x)): x must be in the domain of g, and g(x) in the domain of f.
How to use this function operations test answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
All Precalculus chapters: Precalculus math.
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