These 24 problems practice function notation, domain in interval notation, composition and decomposition, and inverse functions (F-IF.A.1, F-BF.A.1c, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d). They move from evaluating and finding domains to inverses read from graphs and tables.
The last six problems use real contexts and multi-step reasoning, including restricted domains and the inverse of a composition. Show every step.
Problem 1: Evaluate function notation
Let \(f(x)=3x^{2}-2x+1\). Evaluate each expression.
- \(f(3)\)
- \(f(-2)\)
- \(f(a+1)\)
Problem 2: Domain with square roots
Write the domain of each function in interval notation.
- \(f(x)=\sqrt{2x-6}\)
- \(g(x)=\sqrt{10-5x}\)
Problem 3: Domain with denominators
Write the domain of each function in interval notation.
- \(f(x)=\frac{x+1}{x^{2}-9}\)
- \(g(x)=\frac{1}{x^{2}+4}\)
Problem 4: Domain with several restrictions
Write the domain of each function in interval notation.
- \(f(x)=\frac{\sqrt{x+4}}{x-2}\)
- \(g(x)=\ln(5-x)\)
Problem 5: Domain and range from a graph
The graph shows a function \(f\) with a closed point at \((-1,0)\) and an open point at \((3,0)\).
- State the domain in interval notation.
- State the range in interval notation.
- Find every \(x\) with \(f(x)=3\).
Problem 6: Composition from tables
Use the tables to evaluate each expression.
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| f(x) | 2 | 4 | 1 | 3 |
| g(x) | 3 | 1 | 4 | 2 |
- \(f(g(1))\)
- \(g(f(3))\)
- \(f(f(1))\)
- \(g(g(4))\)
Problem 7: Composition of polynomials
Let \(f(x)=x^{2}+1\) and \(g(x)=3x-2\).
- Write and simplify \(f(g(x))\).
- Write and simplify \(g(f(x))\).
- Evaluate \(f(g(2))\).
Problem 8: Composition from a mapping diagram
Function \(g\) maps set A to set B and \(f\) maps set B to set C, as shown.
- Find \(f(g(1))\), \(f(g(2))\) and \(f(g(3))\).
- Write rules for \(g(x)\) and \(f(x)\), then a formula for \(f(g(x))\).
Problem 9: Domain of a composition (roots)
Let \(f(x)=\sqrt{x}\) and \(g(x)=x-5\). Write each composition and state its domain.
- \(f(g(x))\)
- \(g(f(x))\)
Problem 10: Domain of a composition (rational)
Let \(f(x)=\frac{1}{x}\) and \(g(x)=x+2\). Write each composition and state its domain.
- \(f(g(x))\)
- \(g(f(x))\)
Problem 11: Decompose a function
For each \(h(x)\), write \(f(x)\) and \(g(x)\) (neither equal to \(x\)) so that \(h(x)=f(g(x))\).
- \(h(x)=(2x+5)^{4}\)
- \(h(x)=\frac{1}{(x-3)^{2}}\)
- \(h(x)=\sqrt{x^{2}+9}\)
Problem 12: Inverse of a linear function
Find the inverse function of each.
- \(f(x)=5x-3\)
- \(g(x)=-\frac{2}{3}x+4\)
Problem 13: Inverse of a rational function
Let \(f(x)=\frac{2x+1}{x-3}\). Find \(f^{-1}(x)\), then state the domain and range of \(f\).
Problem 14: Verify inverse functions
Decide whether \(f\) and \(g\) are inverse functions by computing both compositions.
- \(f(x)=x^{3}+4\) and \(g(x)=\sqrt[3]{x-4}\)
- \(f(x)=\frac{1}{2}x-3\) and \(g(x)=2x+3\)
Problem 15: Inverse values from a graph
The graph shows a one-to-one function \(f\) through the points \((-2,-4)\), \((0,-1)\), \((2,1)\), \((4,3)\) and \((6,7)\).
- Evaluate \(f(2)\).
- Evaluate \(f^{-1}(3)\).
- Evaluate \(f^{-1}(-1)\).
- Evaluate \(f^{-1}(7)\).
Problem 16: Inverse values from tables
Table 1 shows a function \(f\). Table 2 shows a function \(g\).
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| f(x) | 5 | 3 | 0 | -2 | -6 |
| x | 1 | 2 | 3 |
|---|---|---|---|
| g(x) | 4 | 7 | 4 |
- Find \(f^{-1}(0)\) and \(f^{-1}(-2)\).
- Evaluate \(f^{-1}(f(1))\).
- Explain why \(g\) has no inverse function.
Problem 17: Horizontal line test
Use the horizontal line test on each graph. State which graphs are functions that have an inverse function.
Problem 18: Restrict a domain: vertex form
The graph shows \(f(x)=(x-3)^{2}\).
- Explain why \(f\) has no inverse function on all real numbers.
- Restrict the domain to \(x \ge 3\). Find \(f^{-1}(x)\), with its domain and range.
- Evaluate \(f^{-1}(16)\).
Problem 19: Restrict a domain: complete the square
Let \(f(x)=x^{2}+4x\) with the domain restricted to \(x \ge -2\). Find \(f^{-1}(x)\) and state its domain and range.
Problem 20: Temperature conversion
The function \(C(F)=\frac{5}{9}(F-32)\) converts degrees Fahrenheit to degrees Celsius.
- Find the inverse function \(F(C)\).
- Normal body temperature is \(37^{\circ}\text{C}\). Convert it to Fahrenheit.
- Show that \(C(F(C))=C\).
Problem 21: Coupon and discount order
Maya buys a jacket that costs \(p\) dollars. The store takes 20% off, \(d(p)=0.8p\), and she also has a $5 coupon, \(c(p)=p-5\). The jacket costs $60.
- Find \(c(d(60))\) and \(d(c(60))\).
- Write formulas for \(c(d(p))\) and \(d(c(p))\).
- Explain which order saves more and by how much.
Problem 22: Taxi fare inverse
A taxi charges \(F(m)=2.50+1.80m\) dollars for a ride of \(m\) miles.
- Find the inverse function \(m(F)\) and explain what it computes.
- Jamal pays $20.50. How many miles did he ride?
- Aisha pays $11.50. How many miles did she ride?
Problem 23: Growing ripple
A circular ripple has radius \(r(t)=3t\) feet after \(t\) seconds. Its area is \(A(r)=\pi r^{2}\) square feet.
- Write \(A(r(t))\) in simplest form.
- Find the area after 4 seconds, to the nearest tenth.
- At what time is the area \(81\pi\) square feet?
Problem 24: Inverse of a composition
Let \(f(x)=2x+1\) and \(g(x)=x-4\).
- Find \(f(g(x))\) and its inverse function.
- Find \(f^{-1}(x)\) and \(g^{-1}(x)\), then simplify \(g^{-1}(f^{-1}(x))\).
- Compare your results from (a) and (b).
See every chapter on the Precalculus math page.