Graph the function by using transformation. Find the domain and the range of the function. f (x)= Vx-3 + 1
Graph the function by using transformation. Find the domain and the range of the function. f (x)= Vx-3 + 1
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Concept explainers
Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
![### Analyzing and Graphing the Function
#### Problem Statement:
Graph the function by using transformation. Find the domain and the range of the function.
\[ f(x) = \sqrt{x - 3} + 1 \]
#### Step-by-Step Solution:
1. **Understanding the Base Function:**
The base function here is \( \sqrt{x} \), which is a square root function. The basic graph of \( \sqrt{x} \) starts at the origin (0,0) and increases slowly as x increases, forming a curve.
2. **Applying the Transformation:**
- The given function is \( f(x) = \sqrt{x - 3} + 1 \).
- The expression inside the square root, \( x - 3 \), indicates a horizontal shift.
- Specifically, \( x - 3 \) shifts the graph 3 units to the right.
- The \( +1 \) outside the square root affects the vertical position of the graph, shifting it 1 unit up.
3. **Graphing the Transformed Function:**
- Start by shifting the graph of \( \sqrt{x} \) 3 units to the right.
- Then shift the resulting graph 1 unit upwards.
- Plot some key points for accuracy. For instance, at \( x = 3 \), \( f(3) = \sqrt{3 - 3} + 1 = 0 + 1 = 1 \).
- Continue plotting points and draw the smooth curve that results from connecting them.
4. **Determining the Domain:**
- The domain of a function includes all the possible x-values.
- Since we have a square root function starting from \( x = 3 \), the values under the square root must be non-negative.
- Therefore, \( x - 3 \geq 0 \implies x \geq 3 \).
- **Domain:** \( [3, \infty) \)
5. **Determining the Range:**
- The range of a function includes all the possible y-values.
- For the transformed function \( f(x) \), when \( x = 3 \), \( f(3) = 1 \).
- As \( x \) increases, \( f(x) \) also increases because the square root function always yields non-negative results.
- Thus,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f9e96de-8215-4532-a7d0-6f1fdf87f241%2Fb3706752-504d-40a5-b1c2-4e1776e1879a%2Falyxsrs_processed.png&w=3840&q=75)
Transcribed Image Text:### Analyzing and Graphing the Function
#### Problem Statement:
Graph the function by using transformation. Find the domain and the range of the function.
\[ f(x) = \sqrt{x - 3} + 1 \]
#### Step-by-Step Solution:
1. **Understanding the Base Function:**
The base function here is \( \sqrt{x} \), which is a square root function. The basic graph of \( \sqrt{x} \) starts at the origin (0,0) and increases slowly as x increases, forming a curve.
2. **Applying the Transformation:**
- The given function is \( f(x) = \sqrt{x - 3} + 1 \).
- The expression inside the square root, \( x - 3 \), indicates a horizontal shift.
- Specifically, \( x - 3 \) shifts the graph 3 units to the right.
- The \( +1 \) outside the square root affects the vertical position of the graph, shifting it 1 unit up.
3. **Graphing the Transformed Function:**
- Start by shifting the graph of \( \sqrt{x} \) 3 units to the right.
- Then shift the resulting graph 1 unit upwards.
- Plot some key points for accuracy. For instance, at \( x = 3 \), \( f(3) = \sqrt{3 - 3} + 1 = 0 + 1 = 1 \).
- Continue plotting points and draw the smooth curve that results from connecting them.
4. **Determining the Domain:**
- The domain of a function includes all the possible x-values.
- Since we have a square root function starting from \( x = 3 \), the values under the square root must be non-negative.
- Therefore, \( x - 3 \geq 0 \implies x \geq 3 \).
- **Domain:** \( [3, \infty) \)
5. **Determining the Range:**
- The range of a function includes all the possible y-values.
- For the transformed function \( f(x) \), when \( x = 3 \), \( f(3) = 1 \).
- As \( x \) increases, \( f(x) \) also increases because the square root function always yields non-negative results.
- Thus,
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