Math Problem Statement

Solve c

Solve part (c) of the problem: Given that g(x) = 2f(x - 1), find the domain and range of g.

Solution

To solve part (c) of the problem, we are tasked with finding the domain and range of g(x)=2f(x1)g(x) = 2f(x - 1). Here’s a detailed breakdown:


1. Understanding the Problem

We are given g(x)=2f(x1)g(x) = 2f(x - 1). To determine the domain and range of gg, let’s analyze:

  1. Domain: The values of xx for which g(x)g(x) is defined.
  2. Range: The values of g(x)g(x) as xx varies over its domain.

The graph of f(x)f(x) is defined over 3x5-3 \leq x \leq 5 as given.


2. Domain of g(x)g(x)

  • The term f(x1)f(x - 1) means that the graph of ff is shifted 1 unit to the right.
  • To ensure f(x1)f(x - 1) is defined, the argument x1x - 1 must be within the domain of f(x)f(x), which is 3x5-3 \leq x \leq 5.

Thus, solving for xx: 3x15-3 \leq x - 1 \leq 5 Add 1 to all sides: 2x6-2 \leq x \leq 6

Therefore, the domain of g(x)g(x) is: 2x6\boxed{-2 \leq x \leq 6}


3. Range of g(x)g(x)

The range of g(x)g(x) depends on the range of f(x)f(x), scaled by 2.

  • From the graph of f(x)f(x), the range of ff is 3f(x)4-3 \leq f(x) \leq 4.
  • Since g(x)=2f(x1)g(x) = 2f(x - 1), the range of g(x)g(x) is obtained by multiplying the range of ff by 2.

Scaling the range 3f(x)4-3 \leq f(x) \leq 4 by 2: 2(3)2f(x)2(4)2(-3) \leq 2f(x) \leq 2(4) Simplifying: 6g(x)8-6 \leq g(x) \leq 8

Thus, the range of g(x)g(x) is: 6g(x)8\boxed{-6 \leq g(x) \leq 8}


Final Answer:

  1. Domain of g(x)g(x): 2x6-2 \leq x \leq 6
  2. Range of g(x)g(x): 6g(x)8-6 \leq g(x) \leq 8

Let me know if you need further clarifications! Here are five related questions for practice:

  1. How does shifting f(x)f(x) affect its domain and range?
  2. If g(x)=kf(xh)g(x) = kf(x - h), how would kk affect the range of gg?
  3. Determine the range of h(x)=3f(x+2)h(x) = -3f(x + 2) for the same function ff.
  4. What transformations are applied to f(x)f(x) to obtain g(x)=2f(x1)g(x) = 2f(x - 1)?
  5. If f(x)f(x) is reflected over the xx-axis, how would its range change?

Tip: Scaling a function y=f(x)y = f(x) by a constant kk multiplies the range values by kk.

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Math Problem Analysis

Mathematical Concepts

Functions
Domain and Range
Transformations of Functions

Formulas

Domain shift: If g(x) = f(x - h), domain shifts right by h.
Range scaling: If g(x) = kf(x), range is scaled by k.

Theorems

Function transformations and scaling

Suitable Grade Level

Grades 10-12