Solve values, compositions, and inverses with confidence fast. Review tables, graphs, and related outputs instantly. Master notation using flexible inputs and clear classroom outputs.
Supported syntax includes + - * / ^ ( ), constants pi and e, and functions like sin(x), sqrt(x), ln(x), and abs(x).
This sample shows how the tool handles substitution, compositions, and a linear inverse in one pass.
| f(x) | g(x) | x | Second x | h | Target y | Key outputs |
|---|---|---|---|---|---|---|
| 3*x+2 | x-1 | 4 | 7 | 0.5 | 20 | f(4)=14, g(4)=3, (f∘g)(4)=11, (g∘f)(4)=13, f(4.5)=15.5, inverse x=6 |
| 2*x^2+1 | x+2 | 3 | 5 | 1 | 25 | f(3)=19, g(3)=5, (f∘g)(3)=51, (g∘f)(3)=21, average rate=16 |
Direct substitution: Replace x with the chosen value. For example, f(a) means evaluating the rule at x = a.
Composition: (f ∘ g)(x) = f(g(x)) and (g ∘ f)(x) = g(f(x)). The output of one rule becomes the input of the other.
Average rate of change: [f(b) - f(a)] / (b - a). This measures how quickly the function changes between two selected x values.
Difference quotient: [f(x+h) - f(x)] / h. This approximates instantaneous change when h is small.
Linear inverse: When f(x) = a*x + b, solve y = a*x + b for x, giving x = (y - b) / a.
Point table and graph: The calculator evaluates the entered rule across the selected domain to generate rows for plotting and exporting.
1. Enter the main rule in the f(x) field using x as the variable.
2. Add g(x) if you want function compositions or a second comparison curve.
3. Supply one or two x values. The first is used for direct substitution and compositions. The second helps compute change between points.
4. Enter h for f(x+h) and the difference quotient.
5. Add a target y value if you want the solver to test whether a linear inverse can be produced.
6. Set graph start, graph end, and graph step to control the plotted domain and generated table.
7. Click Solve Function Notation. The result block appears below the header and above the form.
8. Use the CSV and PDF buttons to export the summary and computed point table for classwork, reports, or revision.
You can use numbers, x, parentheses, powers, and standard operators. The solver also supports functions such as sin, cos, tan, sqrt, abs, ln, log, exp, floor, ceil, and round.
Yes. Enter both f(x) and g(x). The solver computes f(g(x)) and g(f(x)) using the first x value, then displays those outputs in the results table.
The inverse section appears when the entered primary rule behaves like a linear function. Nonlinear rules usually do not have a single inverse across all real inputs.
Domain errors occur when a function leaves the real number system. Common examples include dividing by zero, taking the square root of a negative number, or using logarithms with nonpositive values.
The difference quotient estimates how rapidly the function changes near a chosen x value. It is also a common algebra step before learning derivative definitions in calculus.
If the requested step would generate too many points, the tool enlarges the interval slightly. This keeps the graph responsive and prevents oversized exports or slow page rendering.
The primary function still works normally. You will receive direct substitution, rates of change, graphing, and the point table, but composition outputs remain unavailable.
Yes. It is useful for homework checks, revision practice, classroom demonstrations, and quick comparison between rules. The exported table and PDF summary also help with handouts.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.