Calculator Form
Choose function families, enter parameters, then evaluate notation, compositions, tables, and graphs in one place.
Example data table
This sample uses f(x) = 2x + 3 and g(x) = x² - 1 so you can see notation and composition in action.
| x | f(x) | g(x) | f(g(x)) | g(f(x)) |
|---|---|---|---|---|
| -1 | 1 | 0 | 3 | 0 |
| 0 | 3 | -1 | 1 | 8 |
| 2 | 7 | 3 | 9 | 48 |
| 4 | 11 | 15 | 33 | 120 |
Formula used
Function notation basics
f(a) means the output of function f when the input is a.
g(a) works the same way for the second function.
f(g(x)) means evaluate g(x) first, then place that result into f.
g(f(x)) reverses the order, so the outputs can differ.
Average rate of change
Average rate of change measures how fast a function changes between two inputs.
Average rate = [f(x₂) - f(x₁)] / [x₂ - x₁]
Common families
Linear: ax + b
Quadratic: ax² + bx + c
Cubic: ax³ + bx² + cx + d
Power: a(x - h)^b + k
Shifted growth models
Exponential: a·b^(x - h) + k
Logarithmic: a·log_b(x - h) + k
Rational model
Rational: a / (x - h) + k
How to use this calculator
- Choose a family for f(x) and enter its parameters.
- Choose a family for g(x) if you want compositions.
- Enter a primary input x to evaluate notation such as f(x) and g(x).
- Enter a second input to compare outputs and find average rate of change.
- Set the graph start, end, and step values.
- Press Calculate Functions to place the results above the form.
- Review the graph, the summary cards, and the generated value table.
- Use the export buttons to download the results as CSV or PDF.
FAQs
1. What does function notation mean?
Function notation shows how an input produces an output. For example, f(3) means substitute 3 into the rule for f and simplify to get the resulting value.
2. Why do f(g(x)) and g(f(x)) usually differ?
Composition depends on order. The inside function changes the input before the outside function acts, so reversing the order often creates a different output.
3. What happens when a value is undefined?
Undefined results appear when an input breaks a domain rule, such as division by zero or taking a logarithm of a nonpositive number. The table and graph skip those points.
4. Can this page help with domain and range?
Yes. The result cards summarize the expected domain and range for each selected family, including shifts and asymptotes when those features matter.
5. What is average rate of change?
It measures how much the output changes per input unit between two x-values. It is calculated by dividing the output difference by the input difference.
6. Which function types are included?
This calculator supports linear, quadratic, cubic, power, exponential, logarithmic, and rational models, giving you broad coverage for common algebra and precalculus exercises.
7. Can I find an inverse function here?
Yes, when a simple inverse is practical. Some families need domain restrictions first, so the page explains when the inverse is direct and when it needs caution.
8. Why use both a graph and a table?
The graph reveals shape, intercepts, and asymptotes quickly, while the table gives exact sampled outputs. Together, they make function behavior easier to understand.