A table of domain and range of common and useful functions is presented.
Also a Step by Step Calculator to Find Domain of a Function
and a Step by Step Calculator to Find Range of a Function are included in this website.
| Function | Domain | Range |
|---|---|---|
| \( f(x) = x \) | \( (-\infty, +\infty) \) | \( (-\infty, +\infty) \) |
| \( f(x) = x^2 \) | \( (-\infty, +\infty) \) | \( [0, +\infty) \) |
| \( f(x) = x^3 \) | \( (-\infty, +\infty) \) | \( (-\infty, +\infty) \) |
| \( f(x) = x^n \), \( n \) even | \( (-\infty, +\infty) \) | \( [0, +\infty) \) |
| \( f(x) = x^n \), \( n \) odd | \( (-\infty, +\infty) \) | \( (-\infty, +\infty) \) |
| \( f(x) = |x| \) | \( (-\infty, +\infty) \) | \( [0, +\infty) \) |
| \( f(x) = \sqrt{x} \) | \( [0, +\infty) \) | \( [0, +\infty) \) |
| \( f(x) = \sqrt[3]{x} \) | \( (-\infty, +\infty) \) | \( (-\infty, +\infty) \) |
| Function | Domain | Range |
|---|---|---|
| \( f(x) = \sin(x) \) | \( (-\infty, +\infty) \) | \( [-1, 1] \) |
| \( f(x) = \cos(x) \) | \( (-\infty, +\infty) \) | \( [-1, 1] \) |
| \( f(x) = \tan(x) \) | All real numbers except \( \frac{\pi}{2} + n\pi \) | \( (-\infty, +\infty) \) |
| \( f(x) = \sec(x) \) | All real numbers except \( \frac{\pi}{2} + n\pi \) | \( (-\infty, -1] \cup [1, +\infty) \) |
| \( f(x) = \csc(x) \) | All real numbers except \( n\pi \) | \( (-\infty, -1] \cup [1, +\infty) \) |
| \( f(x) = \cot(x) \) | All real numbers except \( n\pi \) | \( (-\infty, +\infty) \) |
| Function | Domain | Range |
|---|---|---|
| \( f(x) = \sin^{-1}(x) \) | \( [-1, 1] \) | \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) |
| \( f(x) = \cos^{-1}(x) \) | \( [-1, 1] \) | \( [0, \pi] \) |
| \( f(x) = \tan^{-1}(x) \) | \( (-\infty, +\infty) \) | \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) |
| \( f(x) = \sec^{-1}(x) \) | \( (-\infty, -1] \cup [1, +\infty) \) | \( [0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2}) \) |
| \( f(x) = \csc^{-1}(x) \) | \( (-\infty, -1] \cup [1, +\infty) \) | \( (-\pi, -\frac{\pi}{2}] \cup (0, \frac{\pi}{2}] \) |
| \( f(x) = \cot^{-1}(x) \) | \( (-\infty, +\infty) \) | \( (0, \pi) \) |
| Function | Domain | Range |
|---|---|---|
| \( f(x) = a^x \) | \( (-\infty, +\infty) \) | \( (0, +\infty) \) |
| \( f(x) = \log_a(x) \) | \( (0, +\infty) \) | \( (-\infty, +\infty) \) |
| \( f(x) = a^x + k \), \( k \) constant | \( (-\infty, +\infty) \) | \( (k, +\infty) \) |
| \( f(x) = \log_a(x - k) \), \( k \) constant | \( (k, +\infty) \) | \( (-\infty, +\infty) \) |
| Function | Domain | Range |
|---|---|---|
| \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) | \( (-\infty, +\infty) \) | \( (-\infty, +\infty) \) |
| \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) | \( (-\infty, +\infty) \) | \( [1, +\infty) \) |
| \( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) | \( (-\infty, +\infty) \) | \( (-1, 1) \) |
| \( \coth(x) = \frac{e^x + e^{-x}}{e^x - e^{-x}} \) | \( (-\infty, 0) \cup (0, +\infty) \) | \( (-\infty, -1) \cup (1, +\infty) \) |
| \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \) | \( (-\infty, +\infty) \) | \( (0, 1) \) |
| \( \operatorname{csch}(x) = \frac{2}{e^x - e^{-x}} \) | \( (-\infty, 0) \cup (0, +\infty) \) | \( (-\infty, 0) \cup (0, +\infty) \) |