Scotch Egg Home
Unit 7: Exponential and General Functions

Topic 4

Domain, Range, and Restrictions

Every function has a domain (the set of all valid inputs xx) and a range (the set of resulting outputs yy).
- The domain includes all xx-values that make the function defined (no division by zero, no square roots of negatives for real numbers). - The range includes all possible yy-values the function can produce.
SAT questions often test your ability to find where a function is undefined or identify valid intervals of input.
f(x)=1x−2is undefined at x=2,f(x) = \frac{1}{x - 2} \quad \text{is undefined at } x = 2,
so its domain is x≠2.x \neq 2.
For square roots:
f(x)=x−3requires x−3≥0,f(x) = \sqrt{x - 3} \quad \text{requires } x - 3 \ge 0,
so domain is x≥3.x \ge 3.
The range depends on the resulting values of f(x)f(x):
If f(x)=x−3, then f(x)≥0.\text{If } f(x) = \sqrt{x - 3}, \text{ then } f(x) \ge 0.

Core Skills

  • Determine domain restrictions from denominators and even roots.
  • Identify the range based on the output behavior.
  • Describe domain/range in set notation or inequality form.
  • Recognize asymptotes or endpoints in graphs.
  • Apply restrictions when combining functions.

Example 1: Rational Function Restriction

f(x)=2x+1x−3f(x) = \frac{2x + 1}{x - 3}
Denominator x−3=0x - 3 = 0 when x=3.x = 3. So x=3x = 3 is not allowed.
Domain: x≠3\boxed{\text{Domain: } x \neq 3}
Range: All real yy except y=2y = 2 (horizontal asymptote).

Example 2: Square Root Function

f(x)=x+4f(x) = \sqrt{x + 4}
Inside the root must be nonnegative:
x+4≥0⇒x≥−4.x + 4 \ge 0 \Rightarrow x \ge -4.
Domain: x≥−4,Range: y≥0\boxed{\text{Domain: } x \ge -4, \quad \text{Range: } y \ge 0}

Example 3: Combined Function Restriction

f(x)=x−2x−5f(x) = \frac{\sqrt{x - 2}}{x - 5}
Root requires x−2≥0⇒x≥2x - 2 \ge 0 \Rightarrow x \ge 2. Denominator requires x≠5x \neq 5.
Domain: x≥2,  x≠5\boxed{\text{Domain: } x \ge 2, \; x \neq 5}

Example 4: Absolute Value Function

f(x)=∣x−3∣f(x) = |x - 3|
Absolute value is always defined, so:
Domain: all real numbers, Range: y≥0.\boxed{\text{Domain: all real numbers, Range: } y \ge 0.}
Graphically, it forms a “V” shape with vertex at (3,0)(3,0).

Example 5: Quadratic Function

f(x)=(x−1)2+2f(x) = (x - 1)^2 + 2
This is always defined (domain: all real). The smallest value occurs when x=1x = 1:
f(1)=(1−1)2+2=2.f(1) = (1 - 1)^2 + 2 = 2.
Domain: all real x,  Range: y≥2.\boxed{\text{Domain: all real } x, \; \text{Range: } y \ge 2.}
Graph has vertex at (1,2)(1, 2) and opens upward.

Example 6: Piecewise Function

f(x)={x+2,x<0x2,x≥0f(x) = \begin{cases} x + 2, & x < 0 \\ x^2, & x \ge 0 \end{cases}
Domain: all real (each piece covers all xx). Range: for x<0x < 0, x+2x+2 covers (−∞,2)(-\infty, 2); for x≥0x \ge 0, x2x^2 covers [0,∞)[0, \infty). Together these two intervals cover every real number.
Domain: all real, Range: all real numbers.\boxed{\text{Domain: all real, Range: all real numbers.}}

Key Takeaways

  • Denominators cannot equal zero.
  • Even roots (square roots) must have nonnegative insides.
  • Domains tell where the function exists; ranges tell possible outputs.
  • Graph endpoints and asymptotes reveal domain/range visually.
  • Always check both when simplifying or combining functions.