range of a function calculator

What is the range of a function?

The range is the set of all possible output values of a function. If you know the domain (the allowed x-values), the range tells you which y-values the function can actually produce.

For example, if f(x) = x² on the interval [-2, 3], the smallest output is 0 (at x = 0) and the largest output is 9 (at x = 3). So the range is [0, 9].

How this range calculator works

This calculator uses a numerical approach:

• It evaluates your function at many evenly spaced points across your chosen interval.
• It tracks the smallest and largest finite outputs found.
• It reports an estimated range and where the estimated min/max occur.

Because this is numerical sampling, results are approximate. Increase sample points for better precision, especially with sharp turns or oscillating functions.

Supported input format

Common expressions

• x^3 - 2*x + 1
• sin(x) + cos(2*x)
• sqrt(x + 4)
• abs(x - 3)
• log(x) (natural logarithm)
• exp(x)

Important notes

• Use * explicitly for multiplication.
• ^ is accepted and converted to exponentiation internally.
• If your function has discontinuities (like 1/(x-2)), the displayed range is based on finite sampled values only.

Examples you can try

1) Quadratic

f(x)=x^2-4x+5 on [-10,10]. This parabola opens upward, so the minimum occurs at its vertex and maximum appears at an endpoint over a closed interval.

2) Trigonometric

f(x)=sin(x) on [0,2π]. Expected range is close to [-1,1].

3) Rational function

f(x)=1/(x-2) on [0,4]. There is a vertical asymptote at x=2, so values explode near that point. Numerical outputs will reflect very large magnitudes depending on sample resolution.

When to use a symbolic method instead

For exact answers, especially in coursework, you may need calculus and algebra:

• Find critical points via derivatives.
• Check interval endpoints.
• Analyze domain restrictions and asymptotic behavior.

This calculator is ideal for exploration, checking intuition, and fast estimates.