F(x/b)

Definition

The expression f(x/b) represents a transformation of a function f(x) where the input variable x is divided by a constant b. This type of transformation is known as a horizontal stretch or compression, depending on the value of b. It affects the domain and range of the original function.

5 Must Know Facts For Your Next Test

1. When b > 1, the function f(x/b) is a horizontal stretch of the original function f(x), with the graph being compressed along the x-axis.
2. When 0 < b < 1, the function f(x/b) is a horizontal compression of the original function f(x), with the graph being stretched along the x-axis.
3. The domain of f(x/b) is the set of all x values for which x/b is in the domain of f(x).
4. The range of f(x/b) is the same as the range of f(x), as the vertical scaling of the function is not affected by this transformation.
5. Horizontal stretches and compressions can be used to model real-world situations, such as the relationship between time and distance in motion problems.

Review Questions

• Explain how the value of the constant b affects the transformation of the function f(x) to f(x/b).
  • The value of the constant b determines whether the function f(x/b) undergoes a horizontal stretch or compression. When b > 1, the function is stretched horizontally, and when 0 < b < 1, the function is compressed horizontally. The domain of the transformed function f(x/b) is affected by this change, as the input variable x is divided by b, but the range remains the same as the original function f(x).
• Describe the relationship between the domain and range of the original function f(x) and the transformed function f(x/b).
  • The domain of the transformed function f(x/b) is the set of all x values for which x/b is in the domain of the original function f(x). This means that the domain of f(x/b) may be different from the domain of f(x), depending on the value of b. However, the range of the transformed function f(x/b) is the same as the range of the original function f(x), as the vertical scaling of the function is not affected by this transformation.
• Analyze how the horizontal stretch or compression of the function f(x) to f(x/b) can be used to model real-world situations.
  • The transformation f(x/b) can be used to model various real-world situations that involve a relationship between time and distance. For example, in motion problems, the function f(x) may represent the distance traveled as a function of time, and the transformed function f(x/b) can be used to model the same relationship but with a different time scale, such as when the speed of the object is changed. This transformation allows for the exploration of how changes in the input variable (time) can affect the output variable (distance) in a meaningful way, which can be useful in applications like transportation, physics, and engineering.

Related terms

Horizontal Stretch: A transformation of a function where the input variable is divided by a constant b > 1, causing the graph to stretch horizontally.

Horizontal Compression: A transformation of a function where the input variable is divided by a constant 0 < b < 1, causing the graph to compress horizontally.

Domain and Range: The set of input values (domain) and output values (range) for which a function is defined.