Find the domain and the range of the function.
f left parenthesis x right parenthesis equals StartSet Start 2 By 1 Matrix 1st Row 1st Column one sixth x comma for x less than 0 2nd Row 1st Column x plus 4 comma for x greater than or equals 0 EndMatrixf(x)= 
1
6x, for x<0 	x+4, for x≥0  
Question content area bottom
Part 1
Choose the correct domain.
A.
​(0,infinity∞​)
B.
​(minus−infinity∞​,infinity∞​)
C.
​(minus−infinity∞​,0) and ​(44​,infinity∞​)
D.
​(minus−infinity∞​,0) and ​(0,infinity∞​)
Part 2
Choose the correct range.
A.
​(minus−infinity∞​,infinity∞​)
B.
​[44​,infinity∞​)
C.
​(minus−infinity∞​,0) and ​[44​,infinity∞​)
D.
​(minus−infinity∞​,0] and ​(44​,infinity∞​)

### Part 1: Domain
The function is defined in two pieces:
- \( f(x) = \frac{1}{6}x \) for \( x < 0 \)
- \( f(x) = x + 4 \) for \( x \geq 0 \)

This means the function is defined for **all real numbers** \( x \), since there is no restriction on the values of \( x \). Therefore, the domain is:
- All real numbers, which is \( (-\infty, \infty) \).

The correct answer for **domain** is:
**B. \( (-\infty, \infty) \)**

### Part 2: Range
Let's analyze the output (range) of each piece:
1. For \( x < 0 \), \( f(x) = \frac{1}{6}x \). As \( x \to 0^- \) (approaching 0 from the left), \( f(x) \to 0 \). As \( x \to -\infty \), \( f(x) \to -\infty \). So, the range of this piece is \( (-\infty, 0) \).
2. For \( x \geq 0 \), \( f(x) = x + 4 \). For \( x = 0 \), \( f(x) = 4 \), and as \( x \to \infty \), \( f(x) \to \infty \). Therefore, the range of this piece is \( [4, \infty) \).

The combined range is the union of these two intervals:
- \( (-\infty, 0) \cup [4, \infty) \).

The correct answer for **range** is:
**D. \( (-\infty, 0] \cup [4, \infty) \)**

---

### Follow-up Questions:
1. How do you calculate the range of piecewise functions?
2. What happens to the range if the slopes or constants in each piece are modified?
3. How do you find the domain of a function if it's defined with multiple conditions?
4. Can a function's range be non-continuous like in this case? Why?
5. What is the significance of the boundary value \( x = 0 \) in this function?

### Tip:
When determining the domain and range of a piecewise function, carefully examine each piece and their respective intervals to ensure proper inclusion or exclusion of values at boundaries.

Find the domain and the range of the function. f(x) = { (1/6)x for x < 0, x + 4 for x ≥ 0 }. Part 1: Choose the correct domain. Part 2: Choose the correct range.

Explore how to find the domain and range of a piecewise function involving linear equations: f(x) = (1/6)x for x < 0 and f(x) = x + 4 for x ≥ 0. This step-by-step guide is perfect for students in Grades 9-11 and covers essential concepts in functions and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation