Operations on Functions Graphing Calculator | Add, Subtract, Multiply, Divide & Compose

Operations on Functions - Graphing Calculator

An online graphing calculator to carry out operations on functions. Five operations are supported: addition, subtraction, multiplication, division, and composition.

The calculator accepts two input functions, \( f(x) \) and \( g(x) \). Algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions may be used. Review the basic functions and operators reference for syntax details.

Examples of formulas to copy and paste: x - 2, -2*x^2 + 3*x - 1, sin(x+pi), sin(pi*x), atan(2*x), (1-x)/(x^2-x-2), sinh(x-1).

Important Syntax Note: The multiplication operator * must be used explicitly (e.g., write 2*sin(x) instead of 2sin(x)), and function arguments must be enclosed in parentheses (e.g., sin(x)).

If needed, free graph paper is available.

Interactive Calculator

Enter formulas for functions \( f \) and \( g \) using the small letter x as the variable, then click one of the operation buttons below to plot \( f \) (blue), \( g \) (green), and the resulting function (red).

Hover your mouse over the graph to trace coordinates, or use the Plotly toolbar to zoom, pan, or download as a PNG.

Interactive Tutorials & Explorations

1. Addition of Two Functions

Let \( f(x) = \sqrt{x} \) and \( g(x) = x \). Input both functions and click (f + g)(x).
Explore the graph (red) of \( (f + g)(x) \). Test specific values such as \( x = 1 \): verify that \( f(1) + g(1) = (f+g)(1) \).
Explore the domain of \( f + g \) graphically. Is it the intersection of the domains of \( f \) and \( g \)?

2. Subtraction of Two Functions

Let \( f(x) = x + 1 \) and \( g(x) = x \). Plot \( (f - g)(x) \). Observe the resulting constant graph.
Let \( f(x) = \sqrt{x - 2} \) and \( g(x) = x \). Verify that at \( x = 2 \), \( (f - g)(2) = f(2) - g(2) \). Explore the domain graphically.

3. Multiplication of Two Functions

Let \( f(x) = x - 2 \) and \( g(x) = x \). Compare the zeros of \( f \), \( g \), and \( f \cdot g \).
Let \( f(x) = \sqrt{x + 2} \) and \( g(x) = x \). Check function values at points like \( x = 2 \) and examine the domain of the product function.

4. Division of Two Functions

Let \( f(x) = 1 \) and \( g(x) = x \). Explore the graph of \( f / g \) and analyze what happens at \( x = 0 \) (vertical asymptote).
Let \( f(x) = \sqrt{x} \) and \( g(x) = x - 1 \). Verify evaluations at \( x = 0 \) and \( x = 4 \), and explain why \( (f/g)(x) \) is undefined at \( x = 1 \).

5. Composition of Two Functions

Let \( f(x) = \sqrt{x} \) and \( g(x) = x^2 \). Estimate \( (f \circ g)(2) \) from the red graph, compare with \( g(2) \) (black) and \( f(g(2)) \) (blue).
Definition: The domain of \( f \circ g \) is the set of all \( x \) such that \( g(x) \) is defined and real, and \( f(g(x)) \) is defined and real.

Exercises & Solutions

View Practice Exercises and Detailed Solutions

Exercise 1: Let \( f(x) = \sqrt{1 - x} \) and \( g(x) = x^2 \). Find the domain of \( (f \circ g)(x) \).
Solution: Since the domain of \( g \) is all real numbers, the domain of \( f \circ g \) requires \( 1 - g(x) \ge 0 \implies 1 - x^2 \ge 0 \), yielding interval \( [-1, 1] \).
Exercise 2: Let \( f(x) = 1 - x \) and \( g(x) = \sqrt{x} \). What is the domain of \( f - g \)?
Solution: Intersection of domain of \( f \) (\( (-\infty, +\infty) \)) and domain of \( g \) (\( [0, +\infty) \)), giving interval \( [0, +\infty) \).
Exercise 3: Let \( f(x) = 1 \) and \( g(x) = \sqrt{x} \). Find the domain of \( f / g \).
Solution: Intersection of domains is \( [0, +\infty) \). Excluding \( x = 0 \) due to division by zero gives interval \( (0, +\infty) \).
Exercise 4: Let \( f(x) = \sqrt{1 - x^2} \) and \( g(x) = \sqrt{4 - x^2} \). Find the domain of \( f \cdot g \).
Solution: Domain of \( f \) is \( [-1, 1] \) and domain of \( g \) is \( [-2, 2] \). Their intersection is \( [-1, 1] \).

Operations on Functions - Graphing Calculator

Interactive Calculator

Interactive Tutorials & Explorations

Exercises & Solutions

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