General functions
Evaluating functions worksheets
Evaluate twelve polynomial and exponential functions. Choose numbers, variable expressions or a mixture, with exact answers and substitution steps.
Preview and downloadGeneral functions practice
Function operations and composition worksheets
Add, subtract, multiply, divide and compose polynomial functions. Choose numbers, variables or mixed inputs, with ten questions and a separate worked answer key.
Ready-made English PDF · fixed settings
Customization below makes a separate PDF.
Fixed English ready-made files. Custom settings below apply only to the configured PDF.
Ready to print
Start with a ready-made worksheet
Read the operation and the order of the functions. For pointwise operations, evaluate both functions at the same input. For composition, evaluate the inner function first. Simplify exactly and state any denominator restriction.
Download the practice pages alone, or include a separate answer key or teacher model. Longer activities may use more than one page.
Customize this worksheetPaper sizes and printing help · More general functions activities
Two functions. Five ways to combine them.
Choose addition and subtraction, multiplication and division, composition, or any combination of these groups. Each worksheet has ten questions. Use numbers, variables, or mixed practice with five of each.
For pointwise operations, both functions receive the same input. Composition passes one function's output into the other. The key shows the intermediate values and the simplified result, so you can follow the order of the calculation.
A function quotient is defined only where its original denominator is nonzero. Canceling a factor does not restore an excluded input. For example, x ÷ x simplifies to 1 only when x ≠ 0.
Learning objective
What this activity builds
Combine two functions using the requested operation, simplify exactly and retain the domain restrictions of a quotient.
Worked example
If f(x) = 2x + 1 and g(x) = x² − 3, find (g ∘ f)(x).
g(f(x)) = (2x + 1)² − 3 = 4x² + 4x − 2.
The right-hand function f is the inner function. Substitute its entire output into g. Reversing the order gives f(g(x)) = 2x² − 5, a different result.
Teaching notes
Make the practice count
- Choose one operation group first, then combine groups when learners can distinguish the notation.
- For addition, subtraction, multiplication and division, use the same input in both functions.
- Read (f ∘ g)(x) as f(g(x)): work from the inside out.
- Keep parentheses around negative values and multi-term substituted expressions.
- Reduce a quotient when possible, but keep every restriction from its original denominator.
- Check an expanded polynomial by substituting a small allowed value into both forms.
Make it yours
Customize the practice
Choose your options, update the preview, then download the exact same pages.
Choose the practice settings, update the preview and print the same questions with a separate answer key.
Enlarged pages scroll. PDF size stays the same.
A useful next step
Related activities
Polynomials
Multiplying binomials worksheets
Expand ten products of two binomials. Choose among four expression structures, form every term pair and collect like terms with a separate worked key.
Preview and download
General functions
Average rate of change from tables worksheets
Find average rates from six function tables. Choose 6 or 12 questions, print Letter or A4, and check each result with a separate worked answer key.
Preview and download