Domain and Range
Overview
The domain of a function refers to all the possible input values (x-values) that the function can accept. The range of a function refers to all the possible output values (y-values) that the function can produce based on its domain.
Domain
The domain is determined by considering the values of \(x\) that make the function valid. Some functions may have restrictions that limit the domain. For example, in the function \( f(x) = \frac{1}{x-2} \), the domain excludes \( x = 2 \) because division by zero is undefined.
Range
The range is the set of all possible output values. To determine the range, you often analyze how the function behaves over its domain and what output values are possible for those inputs.
Examples
- For \( f(x) = x^2 \), the domain is all real numbers (\( -\infty < x < \infty \)) because you can square any number. The range is \( y \geq 0 \), since squaring any real number results in a non-negative value.
- For \( f(x) = \sqrt{x} \), the domain is \( x \geq 0 \) because square roots of negative numbers are undefined in the real number system. The range is also \( y \geq 0 \), as the square root of any non-negative number is non-negative.
Practice Questions
- Find the domain and range of the function \( f(x) = \frac{1}{x+3} \).
Solution
The domain excludes \( x = -3 \) because division by zero is undefined. So, the domain is \( x \neq -3 \) or \( (-\infty, -3) \cup (-3, \infty) \). The range is all real numbers since the function can take any value as output.
- Find the domain and range of the function \( f(x) = \sqrt{x-4} \).
Solution
The domain requires that the expression under the square root, \( x - 4 \), be greater than or equal to 0. So, \( x \geq 4 \). The range is \( y \geq 0 \) since square roots are non-negative.
- What is the domain and range of \( f(x) = x^2 + 2x + 1 \)?
Solution
The domain of this quadratic function is all real numbers, \( (-\infty, \infty) \), because you can plug any real number into the function. The range is \( y \geq -1 \), as the vertex of the parabola is at \( y = -1 \), and the graph opens upwards.
- Find the domain and range of \( f(x) = \frac{x}{x^2 - 9} \).
Solution
The domain excludes \( x = 3 \) and \( x = -3 \) because the denominator becomes zero at these points. So, the domain is \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \). The range is more complex, but the function will take all real values except for 0.
- Find the domain and range of \( f(x) = \log(x-2) \).
Solution
The domain requires that \( x - 2 > 0 \), so \( x > 2 \). The range of a logarithmic function is all real numbers (\( -\infty, \infty \)) because logarithms can produce any real number as output.
- What is the domain and range of \( f(x) = \frac{1}{x^2 + 1} \)?
Solution
The domain is all real numbers, \( (-\infty, \infty) \), because the denominator \( x^2 + 1 \) is never zero. The range is \( 0 < y \leq 1 \), since the values of \( f(x) \) are always positive and never exceed 1.
- Determine the domain and range of \( f(x) = \tan(x) \).
Solution
The domain of \( \tan(x) \) excludes values where \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer, because the tangent function is undefined at those points. The range is all real numbers, \( (-\infty, \infty) \), because tangent can take any value.
- Find the domain and range of \( f(x) = 2x + 1 \).
Solution
The domain of this linear function is all real numbers, \( (-\infty, \infty) \), because you can plug any real number into the function. The range is also all real numbers, \( (-\infty, \infty) \), because linear functions can produce any value.
- What is the domain and range of \( f(x) = \frac{1}{x^2 + x + 1} \)?
Solution
The domain is all real numbers, \( (-\infty, \infty) \), because the denominator \( x^2 + x + 1 \) never equals zero. The range is \( 0 < y \leq \frac{1}{3} \), because the minimum value occurs at the vertex of the parabola and is the reciprocal of the maximum value of the denominator.
- Determine the domain and range of \( f(x) = |x| \).
Solution
The domain is all real numbers, \( (-\infty, \infty) \), because you can take the absolute value of any real number. The range is \( y \geq 0 \), as the absolute value function only produces non-negative values.