It is very common in mathematics to represent a function by an equation. This page presents examples and questions, including solutions, on functions defined by equations. Questions related to the domain and range of a function are also included.
The equation \( y = 2x + 1 \) represents a function that assigns exactly one value for \( y \) for each value of \( x \) in the domain. We say \( y \) is a function of \( x \). Here, \( y \) is the dependent variable and \( x \) is the independent variable because \( y \)'s value depends on \( x \).
Consider the function \( y = 2x + 1 \) with the domain \( D = \{ -2, 0, 1, 2 \} \).
a) Calculate the range \( R \) of \( y \) values.
b) Represent the function using a table, ordered pairs, a graph, and Venn diagrams.
a) Calculate the Range
For \( x = -2 \), \( y = 2(-2) + 1 = -3 \)
For \( x = 0 \), \( y = 2(0) + 1 = 1 \)
For \( x = 1 \), \( y = 2(1) + 1 = 3 \)
For \( x = 2 \), \( y = 2(2) + 1 = 5 \)
Therefore, \( R = \{ -3, 1, 3, 5 \} \).
b) Representations
Table:
| \( x \) | \( -2 \) | \( 0 \) | \( 1 \) | \( 2 \) |
| \( y \) | \( -3 \) | \( 1 \) | \( 3 \) | \( 5 \) |
Ordered Pairs:
\( \{ (-2,-3), (0,1), (1,3), (2,5) \} \)
Graph:
Venn Diagram:
The domain \( D \) is the set of all values of the independent variable \( x \) for which the dependent variable \( y \) is defined.
The range \( R \) is the set of all corresponding values of \( y \).
Find the domain for each equation defining \( y \) as a function of \( x \):
a) \( y = 2 - x \)
b) \( 2y - x = 4 \)
c) \( y = \dfrac{1}{x} \)
d) \( y = \sqrt{x} \)
a) \( y = 2 - x \) is defined for all real numbers. Domain: All real numbers.
b) Solving gives \( y = 2 + \frac{1}{2}x \). Domain: All real numbers.
c) \( y = \dfrac{1}{x} \) is undefined at \( x = 0 \). Domain: All real numbers except \( x = 0 \).
d) \( y = \sqrt{x} \) is defined only for \( x \ge 0 \). Domain: All non-negative real numbers.
Find the range of \( y = x^2 + 1 \) for the domain \( D = \{ 0, 1, 6 \} \).
For \( x = 0 \), \( y = 0^2 + 1 = 1 \)
For \( x = 1 \), \( y = 1^2 + 1 = 2 \)
For \( x = 6 \), \( y = 6^2 + 1 = 37 \)
Range: \( R = \{ 1, 2, 37 \} \).
More on domain and range.
Determine which equations represent \( y \) as a function of \( x \):
a) \( x + y = 2 \)
b) \( x + 5 = |y| \)
c) \( \dfrac{y}{x} = -3 \)
d) \( x + y^2 = 25 \)
a) \( y = 2 - x \). One \( y \) per \( x \). Is a function.
b) For a given \( x \) (e.g., \( x=3 \)), \( |y|=8 \) gives \( y=8 \) and \( y=-8 \). Two outputs for one input. Not a function.
c) \( y = -3x \). One \( y \) per \( x \). Is a function.
d) For a given \( x \) (e.g., \( x=16 \)), \( y^2=9 \) gives \( y=3 \) and \( y=-3 \). Two outputs for one input. Not a function.
The equation \( y = -2x + 6 \) can be written using function notation as \( f(x) = -2x + 6 \). Here, \( f \) is the function name, \( x \) is the input, and \( f(x) \) is the output (image of \( x \) by \( f \)).
To evaluate, substitute the input value: \( f(2) = -2(2) + 6 = 2 \).
Find the range of \( y = x^2 - 1 \) for the domain \( D = \{-1, 3, 5\} \).
Which equations represent \( y \) as a function of \( x \)? Explain.
a) \( -3x + y = x + 3 \)
b) \( |x| + 5 = y \)
c) \( \dfrac{y - 2}{x} = -3 \)
d) \( x + |y| = 25 \)
Find the domain of each function.
a) \( -y = x + 2 \)
b) \( 3x = y + 4 \)
c) \( y = \dfrac{1}{x - 2} \)
d) \( y = \dfrac{1}{\sqrt{x}} \)
Evaluate if possible: \( f(0) \), \( f(-2) \), \( f(10) \), \( g(-4) \), \( g(-3) \) given \( f(x) = -6x + 10 \) and \( g(x) = \dfrac{1}{x+3} \).
For \( x = -1 \), \( y = (-1)^2 - 1 = 0 \)
For \( x = 3 \), \( y = 3^2 - 1 = 8 \)
For \( x = 5 \), \( y = 5^2 - 1 = 24 \)
Range: \( R = \{ 0, 8, 24 \} \).
a) \( y = 4x + 3 \). One output per input. Is a function.
b) \( y = |x| + 5 \). One output per input. Is a function.
c) \( y = -3x + 2 \). One output per input. Is a function.
d) For a given \( x \) (e.g., \( x=4 \)), \( |y| = 21 \) gives two outputs (\( y=21, y=-21 \)). Not a function.
a) \( y = -x - 2 \). Domain: All real numbers.
b) \( y = 3x - 4 \). Domain: All real numbers.
c) Undefined at \( x = 2 \). Domain: All real numbers except \( x = 2 \).
d) Requires \( x > 0 \) (positive real numbers). Domain: All positive real numbers.
\( f(0) = -6(0) + 10 = 10 \)
\( f(-2) = -6(-2) + 10 = 22 \)
\( f(10) = -6(10) + 10 = -50 \)
\( g(-4) = \dfrac{1}{-4+3} = -1 \)
\( g(-3) = \dfrac{1}{-3+3} = \text{undefined} \).
Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5
Algebra and Trigonometry with Analytic Geometry - R.E.Larson, et al. - 1997 - ISBN: 0-669-41723-8
Functions in Mathematics
Domain and Range of Functions
Free Online Tutorials on Functions and Algebra