The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range, you need to determine the values that the function can take as its output, depending on the domain (the set of all possible input values or x-values).

Steps to Find the Range of a Function

1. Identify the Function: Start by looking at the function you're dealing with. For example, let's consider a simple quadratic function: $f(x) = x^2 - 4$.

2. Determine the Domain: The domain is the set of all possible input values (x-values) for the function. For the function $f(x) = x^2 - 4$, the domain is all real numbers, $x \in \mathbb{R}$, because any real number can be squared.

3. Analyze the Function's Behavior:

   • Quadratic Function: The function $f(x) = x^2 - 4$ is a parabola opening upwards (since the coefficient of $x^2$ is positive).
   • Vertex: The vertex of the parabola occurs at $x = 0$. Substituting $x = 0$ into the function, we get $f(0) = 0^2 - 4 = -4$.
   • Direction: Since the parabola opens upwards, the value $f(x)$ will be greater than or equal to $-4$.

4. Determine the Range:

   • Based on the analysis, the minimum value of $f(x)$ is $-4$, and because the parabola opens upwards and extends infinitely, the function's range is all values $y \geq -4$.
   • Range: Therefore, the range of $f(x) = x^2 - 4$ is $y \geq -4$, or in interval notation, $\text{Range} = [-4, \infty)$.

Example 2: Find the Range of $g(x) = \frac{1}{x}$

1. Identify the Function: $g(x) = \frac{1}{x}$.

2. Domain: The domain is $x \neq 0$ because division by zero is undefined.

3. Analyze the Behavior:

   • As $x$ approaches 0 from the positive side, $g(x)$ becomes very large (approaches infinity).
   • As $x$ approaches 0 from the negative side, $g(x)$ becomes very large in the negative direction (approaches negative infinity).
   • Therefore, the function never takes the value 0.

4. Determine the Range:

   • Since $g(x)$ can take any value except 0, the range is $y \neq 0$, or in interval notation, $\text{Range} = (-\infty, 0) \cup (0, \infty)$.

Summary

• Step 1: Identify the function.
• Step 2: Determine the domain.
• Step 3: Analyze the function’s behavior.
• Step 4: Determine the range from the function’s behavior.

Questions:

1. How do you find the range of a function if it’s a cubic polynomial?
2. What is the range of $f(x) = \sin(x)$?
3. How would you find the range of a rational function like $f(x) = \frac{2x+3}{x-1}$?
4. What role does the domain play in determining the range of a function?
5. Can a function have a range that is a single value? Give an example.

Tip:

When determining the range of more complex functions, graphing the function can be very helpful in visualizing the possible output values.