Domain and Range of a Function
Understanding the domain and range of a function is crucial in Algebra. The domain refers to all possible inputs for the function, while the range refers to all possible outputs. This lesson will guide you through finding the domain and range for various types of functions.
Outline
- Definition of Domain and Range
- Finding the Domain
- Finding the Range
- Examples and Practice Exercises
- Homework
- Revision
- YouTube Video Explanation
Definition of Domain and Range
The domain of a function is the set of all possible values that can be input into the function. The range is the set of all possible values that the function can output.
Finding the Domain
To find the domain, consider the following:
- For rational functions, the domain excludes values that make the denominator zero.
- For square root functions, the domain includes values that keep the radicand non-negative.
- For logarithmic functions, the domain includes values that keep the argument positive.
Finding the Range
To find the range, analyze how the function behaves and identify the set of possible outputs:
- For polynomial functions, the range can be all real numbers or a specific subset.
- For quadratic functions, the range depends on the vertex of the parabola.
- For rational functions, the range excludes values that make the function undefined.
Examples
Example 1
Find the domain of \( f(x) = \frac{1}{x - 3} \).
Solution: The domain is all real numbers except \( x = 3 \), where the function is undefined.
Example 2
Find the domain of \( f(x) = \sqrt{x + 2} \).
Solution: The domain is \( x \geq -2 \) because the radicand must be non-negative.
Example 3
Find the range of \( f(x) = x^2 - 4 \).
Solution: The range is \( y \geq -4 \) because the minimum value of \( x^2 \) is 0.
Example 4
Find the range of \( f(x) = \frac{2x + 1}{x - 1} \).
Solution: The range is all real numbers except 1, which is the vertical asymptote.
Example 5
Find the domain of \( f(x) = \log(x - 5) \).
Solution: The domain is \( x > 5 \) because the argument of the logarithm must be positive.
Example 6
Find the range of \( f(x) = -x^2 + 2x + 3 \).
Solution: The range is \( y \leq 4 \), where 4 is the maximum value at the vertex of the parabola.
Exercises
Exercise 1
Find the domain of \( f(x) = \frac{5}{x^2 - 9} \).
Solution: The domain is all real numbers except \( x = \pm 3 \), where the denominator is zero.
Exercise 2
Find the domain of \( f(x) = \sqrt{3x - 7} \).
Solution: The domain is \( x \geq \frac{7}{3} \).
Exercise 3
Find the range of \( f(x) = \frac{3x - 4}{2x + 1} \).
Solution: The range is all real numbers except \( \frac{3}{2} \), which is the horizontal asymptote.
Exercise 4
Find the range of \( f(x) = x^3 - 3x \).
Solution: The range is all real numbers because cubic functions have no restrictions on their range.
Exercise 5
Find the domain of \( f(x) = \log(x^2 - 4) \).
Solution: The domain is \( x < -2 \) or \( x > 2 \) because the argument of the logarithm must be positive.
Exercise 6
Find the domain and range of \( f(x) = \frac{1}{x^2 + 1} \).
Solution: The domain is all real numbers. The range is \( 0 < y \leq 1 \).
Homework
Complete the following problems for further practice:
- Find the domain and range for the following functions: \( f(x) = \frac{1}{x^2 - 4} \), \( g(x) = \sqrt{2x + 1} \), and \( h(x) = x^3 - 2x \).
- Create a set of functions and determine their domain and range.
- Discuss real-life scenarios where understanding domain and range is essential.
Revision
Review the key concepts:
- The domain of a function is all possible input values.
- The range of a function is all possible output values.
- Consider the function's behavior and restrictions when finding the domain and range.
Video Explanation
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