To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x^3? What is its range?

Verified step by step guidance

1. Step 1: To identify the graph of the function ƒ(x)=x^3, look for a graph that starts from the bottom left, crosses through the origin (0,0), and continues to the top right. This is because the cube of a negative number is negative, the cube of zero is zero, and the cube of a positive number is positive.
2. Step 2: The graph of ƒ(x)=x^3 is a smooth curve that increases without bound as x increases and decreases without bound as x decreases. It does not have any sharp turns or cusps.
3. Step 3: The range of a function is the set of all possible output values (y-values) that we can get by substituting the x-values into the function. For the function ƒ(x)=x^3, there is no restriction on the values that y can take.
4. Step 4: Since the graph of ƒ(x)=x^3 extends indefinitely in both the positive and negative directions along the y-axis, the range of the function is all real numbers.
5. Step 5: Therefore, the range of the function ƒ(x)=x^3 is (-∞, ∞).

Key Concepts

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form ƒ(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0. The graph of a cubic function has a characteristic 'S' shape, which can have one or two turning points depending on the coefficients. Understanding the general shape and behavior of cubic functions is essential for identifying their graphs.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. For cubic functions, key points include the y-intercept (where x=0) and the behavior as x approaches positive or negative infinity. Recognizing these points helps in sketching the graph accurately and understanding its overall shape and direction.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the cubic function ƒ(x) = x^3, the range is all real numbers, as the function can take any value from negative to positive infinity. Understanding the range is crucial for interpreting the behavior of the function and answering questions related to its outputs.