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

Problem 1: Add and subtract functions

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

1. \((f+g)(x)\)
2. \((f-g)(x)\)

Problem 2: Multiply functions

Let \(f(x)=x+3\) and \(g(x)=2x-5\).

1. Find \((fg)(x)\).
2. Evaluate \((fg)(4)\).

Problem 3: Divide functions and state the domain

Let \(f(x)=x^{2}-9\) and \(g(x)=x-3\). Write \((\frac{f}{g})(x)\) in simplest form and state its domain.

Problem 4: Operations from graphs

The graphs of \(f\) and \(g\) are shown. Read values from the graph to find each result.

1. \((f+g)(1)\)
2. \((f-g)(5)\)
3. \((fg)(3)\)

Problem 5: Evaluate a composition

Let \(f(x)=2x-5\) and \(g(x)=x^{2}+1\). Evaluate each expression.

1. \(f(g(3))\)
2. \(g(f(3))\)

Problem 6: Write composite functions

Let \(f(x)=x+4\) and \(g(x)=3x^{2}\). Write each composite function in simplest form.

1. \(f(g(x))\)
2. \(g(f(x))\)

Problem 7: Trace a value through a machine

Let \(g(x)=3x-1\) and \(f(x)=x^{2}\).

1. Find \(g(2)\) and then \(f(g(2))\).
2. Write \(f(g(x))\) as a polynomial and check it at \(x=2\).

Problem 8: Domain of a composition

Let \(f(x)=\sqrt{x}\) and \(g(x)=x-5\).

1. Write \(f(g(x))\) and state its domain.
2. Write \(g(f(x))\) and state its domain.

Problem 9: Decompose a function

Write each function as \(h(x)=f(g(x))\) by naming an inner function \(g\) and an outer function \(f\), neither equal to \(x\).

1. \(h(x)=(5x-2)^{3}\)
2. \(h(x)=\frac{1}{x^{2}+1}\)

Problem 10: Inverse of a linear function

Find the inverse of \(f(x)=4x-9\) and check it with the point \((5,11)\).

Problem 11: Inverse with fractions

Find the inverse of \(f(x)=-\frac{2}{3}x+4\).

Problem 12: Verify inverses by composition

Show that \(f(x)=5x+3\) and \(g(x)=\frac{x-3}{5}\) are inverse functions.

Problem 13: Are they inverses?

Decide whether \(f(x)=2x+6\) and \(g(x)=\frac{x}{2}-6\) are inverse functions. If not, write the correct inverse of \(f\).

Problem 14: Inverse with a restricted domain

Let \(f(x)=x^{2}+3\) for \(x\ge 0\). Find \(f^{-1}(x)\), state its domain, and evaluate \(f^{-1}(12)\).

Problem 15: Inverse of a square root function

Find the inverse of \(f(x)=\sqrt{x+2}\) and state the domain of \(f^{-1}\).

Problem 16: Reflect points over y = x

The graph shows a linear function \(f\).

1. Give the coordinates of the points on the graph of \(f^{-1}\) that correspond to the four marked points.
2. Write a formula for \(f^{-1}(x)\).

Problem 17: Horizontal line test

Use the graph of \(f(x)=(x-2)^{2}-1\) to answer.

1. Does the horizontal line \(y=3\) meet the graph more than once? Name the points.
2. Does \(f\) have an inverse function on all real numbers? Explain.
3. Name a restriction of the domain that makes \(f\) one-to-one.

Problem 18: Inverse after restricting the domain

Let \(f(x)=(x-2)^{2}-1\) for \(x\ge 2\). Find \(f^{-1}(x)\) and its domain, and check with \(f(5)\).

Problem 19: Inverse of a rational function

Find the inverse of \(f(x)=\frac{2x}{x+3}\) for \(x\neq -3\).

Problem 20: Compositions that are equal

Let \(f(x)=2x-1\) and \(g(x)=x^{2}\). Find all values of \(x\) for which \(f(g(x))=g(f(x))\).

Problem 21: Temperature conversion

Fahrenheit temperature is \(F(C)=\frac{9}{5}C+32\) for Celsius temperature \(C\).

1. Find the inverse function \(C(F)\).
2. Use it to convert \(95^{\circ}\text{F}\) and \(-4^{\circ}\text{F}\) to Celsius.

Problem 22: Car rental inverse

Maya rents a car for a base fee of $25 plus $0.45 per mile, so the cost is \(C(m)=0.45m+25\) dollars for \(m\) miles.

1. Find the inverse function \(m(C)\).
2. Explain what the inverse tells you, then find the miles driven when the bill is $70.

Problem 23: Discount and coupon order

A store takes 20% off every price, and Jamal has a $5-off coupon. Let \(d(x)=0.8x\) be the discount and \(c(x)=x-5\) the coupon.

1. Write \(c(d(x))\) and \(d(c(x))\).
2. Find the cost of a $60 jacket for each order.
3. Which order is cheaper, and by how much for any price?

Problem 24: Composition in context

A circular oil spill spreads so that its radius after \(t\) seconds is \(r(t)=3t\) inches. The area is \(A(r)=\pi r^{2}\) square inches.

1. Write \(A(r(t))\) in simplest form.
2. Find the area after 2 seconds, in terms of \(\pi\) and rounded to the nearest tenth.

See every chapter on the Algebra 2 math page.