Functions: Domain, Range, Composition and Inverses: Precalculus Math Test – PDF Download

Precalculus Functions Review Test: questions 1–5

Question 1: Domain in interval notation (2 points)

1. \(f(x)=\sqrt{3x-12}\) (1 point)
2. \(g(x)=\frac{x}{x^{2}-x-6}\) (1 point)

Question 2: Domain and range from a graph (2 points)

The graph shows a function \(f\) that starts at the point \((-1,3)\).

1. State the domain of \(f\). (1 point)
2. State the range of \(f\). (1 point)

Question 3: Composition of functions (2 points)

Let \(f(x)=x^{2}-3x\) and \(g(x)=x+4\).

1. Write and simplify \(f(g(x))\). (1 point)
2. Evaluate \(g(f(2))\). (1 point)

Question 4: Decompose and find a domain (2 points)

Let \(h(x)=\sqrt{x^{2}-16}\).

1. Write \(f(x)\) and \(g(x)\) so that \(h(x)=f(g(x))\) and neither is \(x\). (1 point)
2. State the domain of \(h\) in interval notation. (1 point)

Question 5: Find and verify an inverse (3 points)

Let \(f(x)=\frac{3x-2}{x+1}\).

1. Find \(f^{-1}(x)\). (2 points)
2. Show that \(f^{-1}(f(x))=x\). (1 point)

Precalculus Functions Review Test: questions 6–9

Question 6: Inverse values from a graph (3 points)

The graph shows a one-to-one function \(f\) through the points \((-4,-3)\), \((-2,0)\), \((0,2)\), \((2,3)\) and \((4,6)\).

1. Find \(f^{-1}(2)\). (1 point)
2. Find \(f^{-1}(6)\). (1 point)
3. State the domain and range of \(f^{-1}\). (1 point)

Question 7: Restrict a domain (3 points)

Let \(f(x)=x^{2}-6x+5\), shown below.

1. Explain why \(f\) has no inverse function as given. (1 point)
2. Restrict the domain to \(x \ge 3\) and find \(f^{-1}(x)\) with its domain. (2 points)

Question 8: Temperature inverse (2 points)

The function \(C(F)=\frac{5}{9}(F-32)\) converts degrees Fahrenheit to degrees Celsius.

1. Find \(F(C)\), the inverse function. (1 point)
2. A freezer is set to \(-15^{\circ}\text{C}\). Find the Fahrenheit temperature. (1 point)

Question 9: Find an unknown function (1 point)

Let \(g(x)=3x-5\) and \(f(g(x))=6x-7\). Find \(f(x)\).

How to use this precalculus functions review test

Take this precalculus functions review test without notes, then grade it with the test answer key out of 20 points.

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