Math Problem Statement

The function f is given by f(x) = 2^x, and the function g is given by g(x) = f(x)/8. For which of the following transformations is the graph of g the image of the graph of f?

Solution

The problem provides two functions:

  1. f(x)=2xf(x) = 2^x
  2. g(x)=f(x)8g(x) = \frac{f(x)}{8}

It asks for the transformation that makes the graph of gg the image of the graph of ff.

To solve this:

  1. Substitute f(x)=2xf(x) = 2^x into g(x)g(x): g(x)=2x8g(x) = \frac{2^x}{8}
  2. Simplify 2x8\frac{2^x}{8}. Since 8=238 = 2^3, this can be written as: g(x)=2x23=2x3g(x) = \frac{2^x}{2^3} = 2^{x - 3}
  3. This simplification shows that g(x)=2x3g(x) = 2^{x - 3}, which represents a horizontal shift of the function f(x)=2xf(x) = 2^x.

Since g(x)=2x3g(x) = 2^{x - 3}, the transformation applied to f(x)f(x) is a horizontal shift 3 units to the right.

Would you like more details or further clarification?

Related Questions

  1. What would be the transformation if g(x)=f(x)+8g(x) = f(x) + 8 instead?
  2. How does a vertical shift affect the function f(x)=2xf(x) = 2^x?
  3. What if g(x)=f(x3)g(x) = f(x - 3) instead? How would that change the graph?
  4. How would you express g(x)=2x5g(x) = 2^{x - 5} in terms of transformations?
  5. What effect does multiplying f(x)f(x) by a constant have on the graph?

Tip

For exponential functions, transformations involving addition or subtraction in the exponent affect the graph horizontally, while multiplications outside the function affect the graph vertically.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Function Transformations

Formulas

f(x) = 2^x
g(x) = f(x)/8
g(x) = 2^(x - 3)

Theorems

Properties of Exponential Functions
Transformation Rules for Functions

Suitable Grade Level

Grades 10-12