Domain and Range - Precalculus & Trigonometry Step by Step

Find the domain and range of a rational function f(x) = 1/(x-5) by identifying values that make the denominator zero.

The range is the set of all possible outputs (y-values). For f(x) = 1/(x - a), the range excludes zero because the numerator is a nonzero constant. Example: f(x) = 1/(x - 5). Set x - 5 = 0 → x = 5 is excluded. Domain: (-∞, 5) ∪ (5, ∞). Range: (-∞, 0) ∪ (0, ∞).

Example

Find excluded x-values from the denominator, then state domain and range.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

f(x)=\frac{ 1 }{x- 5 }

Step 2 — Denominator zero

Set the denominator to zero: x - 5 = 0, so x = 5 is excluded.

x- \hl{5} = 0 \Rightarrow x= \hl{5}

Step 3 — Domain

Domain excludes 5: all real x except 5.

(-\infty, \hl{5} )\cup( \hl{5} ,\infty)

Step 4 — Range

Range excludes 0: all real y except 0.

(-\infty, \hl{0} )\cup( \hl{0} ,\infty)

domain-range The domain of a function is the set of all valid inputs (x-values). For rational functions, exclude values that make the denominator zero.