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Magazine Chapter 3: Problem 347
In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation.$$ f(x)=\frac{1}{2} x+1 $$
Short Answer
Expert verified
Domain: \( (-\infty, \infty) \); Range: \( (-\infty, \infty) \).
Step by step solution
01
- Identify the function
The given function is a linear function: \[ f(x) = \frac{1}{2}x + 1 \].
02
- Determine the slope and y-intercept
For the function \( f(x) = \frac{1}{2}x + 1 \), the slope (m) is \( \frac{1}{2} \) and the y-intercept (b) is 1. This means the line crosses the y-axis at the point (0, 1).
03
- Plot the y-intercept
On a graph, mark the point (0, 1) where the line crosses the y-axis.
04
- Use the slope to find another point
Starting from the y-intercept (0, 1), use the slope \( \frac{1}{2} \). The slope means that for every 2 units moved horizontally, the function moves 1 unit vertically. From (0, 1), move 2 units to the right and 1 unit up to reach the point (2, 2).
05
- Draw the line
Draw a straight line through the points (0, 1) and (2, 2). This is the graph of the function \( f(x) = \frac{1}{2}x + 1 \).
06
- State the domain
The domain of a linear function is all real numbers. In interval notation, it is \( (-\infty, \infty) \).
07
- State the range
The range of a linear function with a non-zero slope is also all real numbers. In interval notation, it is \( (-\infty, \infty) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope
The slope of a line represents how steep the line is. In the linear function \( f(x) = \frac{1}{2}x + 1 \), the slope (often represented by \( m \)) is \( \frac{1}{2} \). This means for every 2 units you move horizontally to the right, the line moves 1 unit vertically up. The slope directly affects the angle of the line compared to the x-axis.
- If the slope is positive, the line rises as it moves from left to right.
- If the slope is negative, the line falls as it moves from left to right.
- If the slope is zero, the line is horizontal, indicating no vertical change regardless of horizontal movement.
y-intercept
The y-intercept is where the line crosses the y-axis. For the function \( f(x) = \frac{1}{2}x + 1 \), the y-intercept \( b \) is 1. This means the line crosses the y-axis at the point (0, 1). Marking the y-intercept on a graph gives you a starting point for drawing the line.
- To find the y-intercept from the equation, set \(x = 0 \) and solve for \( f(0) \).
- It is visually easy to locate as it is a direct intersection with the y-axis.
interval notation
Interval notation is a way of writing subsets of the real number line. It is often used to describe the domain and range of functions. When dealing with linear functions:
- The domain represents all possible input values (x-values).
- The range represents all possible output values (y-values).
- Use brackets \([]\) to denote inclusive endpoints.
- Use parentheses \(()\) to denote exclusive endpoints or for infinities.
domain and range
The domain and range of a function describe the possible values of inputs (x-values) and outputs (y-values), respectively. For linear functions like \( f(x) = \frac{1}{2}x + 1 \), both the domain and range encompass all real numbers.
- The domain: \( (-\infty, \infty) \) indicates that any x-value from negative infinity to positive infinity can be plugged into the function.
- The range: \( (-\infty, \infty) \) indicates that the function can produce any y-value from negative infinity to positive infinity.
