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Magazine Chapter 9: Problem 53
Graph each linear or constant function. Give the domain and range. \(f(x)=-2 x+5\)
Short Answer
Expert verified
Domain: (-鈭�, 鈭�), Range: (-鈭�, 鈭�)
Step by step solution
01
- Identify the function type
The given function is a linear function because it is in the form of a straight line, which is represented by the equation: \[ f(x) = -2x + 5 \]
02
- Create a table of values
Choose some values for \( x \) (e.g., -2, -1, 0, 1, 2) and calculate the corresponding \( f(x) \) values:\( x = -2 \rightarrow f(x) = -2(-2) + 5 = 9 \)\( x = -1 \rightarrow f(x) = -2(-1) + 5 = 7 \)\( x = 0 \rightarrow f(x) = -2(0) + 5 = 5 \)\( x = 1 \rightarrow f(x) = -2(1) + 5 = 3 \)\( x = 2 \rightarrow f(x) = -2(2) + 5 = 1 \)
03
- Plot the points on a Cartesian plane
Using the points calculated in the previous step, plot the points on a graph: \((-2, 9), (-1, 7), (0, 5), (1, 3), (2, 1)\).
04
- Draw the line
Connect the plotted points with a straight line. This line represents the graph of the function \( f(x) = -2x + 5 \).
05
- Identify the domain
For a linear function like \( f(x) = -2x + 5 \), the domain includes all real numbers because the function is defined for every value of \( x \). Hence, the domain is:\[ \text{Domain}: \ (-\infty, \infty) \]
06
- Identify the range
For a linear function, the range also includes all real numbers because the function can produce every possible \( y \) value. Hence, the range is:\[ \text{Range}: \ (-\infty, \infty) \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain
In mathematics, the domain of a function is the set of all possible input values (typically represented as x-values) for which the function is defined. For a linear function such as \(f(x) = -2x + 5\), the domain is all real numbers because you can substitute any real number for x, and the function will produce a corresponding output. This is typically written as:
\[ \text{Domain}: \ (-\infty, \infty) \]. This means that the function works for every possible x-value.
\[ \text{Domain}: \ (-\infty, \infty) \]. This means that the function works for every possible x-value.
Range
The range of a function is the set of all possible output values (typically represented as y-values) that the function can produce. For a linear function such as \(f(x) = -2x + 5\), the range is also all real numbers. This is because multiplying and adding real numbers will give you all possible y-values. Thus, the function can produce any y-value, and we write this as:
\[ \text{Range}: \ (-\infty, \infty) \]. As with the domain, the function is not limited and can create any y-value.
\[ \text{Range}: \ (-\infty, \infty) \]. As with the domain, the function is not limited and can create any y-value.
Plotting Points
Plotting points is a fundamental step in graphing linear functions. To plot a point, you need a pair of x and y values that satisfy the function's equation. For our function \(f(x) = -2x + 5\), you can pick several x-values and then use the equation to find the corresponding f(x) or y-values. For example:
- If x = -2, then f(x) = -2(-2) + 5 = 9, giving us the point (-2, 9).
- If x = 0, then f(x) = -2(0) + 5 = 5, giving us the point (0, 5).
- Repeat this process for a few other x-values to get a good set of points.
Creating a Table of Values
Creating a table of values helps in organizing the x and y values, making it easier to plot points. For our function \(f(x) = -2x + 5\), let鈥檚 use x-values -2, -1, 0, 1, 2. We calculate the corresponding y-values to fill our table:
- For x = -2, y = -2(-2) + 5 = 9
- For x = -1, y = -2(-1) + 5 = 7
- For x = 0, y = -2(0) + 5 = 5
- For x = 1, y = -2(1) + 5 = 3
- For x = 2, y = -2(2) + 5 = 1
Linear Equation
A linear equation is an equation that models a straight line when graphed. It has the general form \(y = mx + b\), where m is the slope (rate of change) and b is the y-intercept (the value of y when x is zero). In our example, \(f(x) = -2x + 5\), the slope m is -2, and the y-intercept b is 5.
- The slope of -2 means the line inclines downward from left to right, decreasing by 2 units for each 1 unit increase in x.
- The y-intercept 5 is where the line crosses the y-axis.
