Page 56 - Algebra

Domain and Range of Functions
Domain is the set of numbers for which the function f(x) is defined.
For example, f(x) = 1 , the domain is R –(0), the function is defined for all real number except for zero.
2𝑥
f(x) = √3𝑥, the domain is defined for all non-negative integers.
REMEMBER:
For finding the domain of an algebraic function
• Ensure that the denominator is never 0.
• Expression under a root should always be positive; it can never be negative.
For logarithmic function
• logab is defined when a > 0, b > 0 and a ≠ 1
For trigonometric functions
• cos x and sin x are defined for all values of x
• tan x and sec x are defined for all values of x except x = (2n + 1)π, n holds true for all integers 2
• sec x and cot x are defined for all values of x except x = nπ, where n holds true for all integers
For exponential functions
• bx is defined for all values of x, where b > 0
Range
Once you calculate the domain of the function, you can find the range of the function using the following steps.
• Calculate the domain of the function y = f(x)
• Solve y = f(x) and find the value of x in terms of y
• Range of the function is all the real values of y for which even x is real
Worked Example
Find the domain and range of √𝑥 − 10
Solution:
The function is defined for x – 10 ≥ 0
x ≥ 10
=y x – 10 = y2
x = y2 + 10 for x > 0
Range of the function [0, ∞)
domain of √𝑥 − 10 is [10, ∞)
Now, equate √𝑥 − 10 to y
√𝑥 − 10
Page 55 of 177
Algebra I & II