Problem 30
Question
In Exercises 29-32, graph the functions \(f\), \(g\), and \(f\) + \(g\) on the same set of coordinate axes. \(f(x) = \frac{1}{3}x\), \(g(x) = -x + 4\)
Step-by-Step Solution
Verified Answer
The function \(f(x) = \frac{1}{3}x\) is a straight line passing through the origin sloping upwards. Alternatively, \(g(x) = -x + 4\) is a line starting from the point \((0,4)\) and sloping downwards on the left. The sum of the functions \(f(x) + g(x) = -\frac{2}{3}x + 4\) is also a straight line starting at the point \((0,4)\) and sloping downwards to the left, but less steeply than \(g(x)\).
1Step 1: Plot \(f(x)\)
Firstly, graph the function \(f(x) = \frac{1}{3}x\). The slope of this line is \(\frac{1}{3}\), meaning for every unit move to the right, the value of \(f\) increases by \(\frac{1}{3}\). When \(x = 0\), \(f(x) = 0\) as well making this a straight line passing through the origin.
2Step 2: Plot \(g(x)\)
Next, graph the function \(g(x) = -x + 4\). The slope of this line is -1, meaning for every unit moved to the right, the value of \(g\) decreases by 1. The function cuts the y-axis at \(y = 4\). This line would therefore start from the point \((0,4)\) and slope downwards towards the left.
3Step 3: Plot \(f(x) + g(x)\)
Finally, graph the sum of the two functions, \(f(x) + g(x)\) which equals \(\frac{1}{3}x -x + 4\). Simplifying this yields \(-\frac{2}{3}x + 4\). For every unit moved to the right, the value of \(f(x) + g(x)\) decreases by \(\frac{2}{3}\). The line crosses the y-axis at \(y = 4\). The line starts at the point \((0,4)\) and slopes downwards towards the left, less steeply than \(g(x)\) due to its smaller gradient.
Key Concepts
Function AdditionSlope of a LineY-InterceptCoordinate Axes
Function Addition
Understanding function addition is crucial when dealing with multiple linear functions. In the context of our exercise, we consider the functions f(x) and g(x), and their sum, f(x) + g(x). To perform the addition, we simply combine the x-terms and constant terms of both functions.
When doing this, we must maintain the operations indicated between the terms. For the given functions, f(x) = \( \frac{1}{3}x \) and g(x) = -x + 4, adding them yields f(x) + g(x) = \( \frac{1}{3}x \) - x + 4. It's a straightforward process but do not skip simplifying the resulting function to its lowest terms, which is essential to make graphing and interpreting the function easier.
When doing this, we must maintain the operations indicated between the terms. For the given functions, f(x) = \( \frac{1}{3}x \) and g(x) = -x + 4, adding them yields f(x) + g(x) = \( \frac{1}{3}x \) - x + 4. It's a straightforward process but do not skip simplifying the resulting function to its lowest terms, which is essential to make graphing and interpreting the function easier.
Slope of a Line
The slope of a line is a numerical measure of its steepness, often thought of as 'rise over run'. For any linear function in the form y = mx + b, m represents the slope. In the first function from our exercise, f(x) = \( \frac{1}{3}x \), the slope is \( \frac{1}{3} \), indicating a gentle incline.
Conversely, g(x) = -x + 4 has a slope of -1, pointing to a steeper decline for every unit increase in x. Understanding the slope is key to graphing the function correctly and predicting its behavior. A positive slope means the line goes upwards as we move from left to right, while a negative slope means just the opposite.
Conversely, g(x) = -x + 4 has a slope of -1, pointing to a steeper decline for every unit increase in x. Understanding the slope is key to graphing the function correctly and predicting its behavior. A positive slope means the line goes upwards as we move from left to right, while a negative slope means just the opposite.
Y-Intercept
Every linear equation also has a y-intercept, which is the point where the line crosses the y-axis. This happens when x is zero. In the equation form y = mx + b, b is the y-intercept.
For function g(x), the y-intercept is 4, as denoted by the equation -x + 4. This means the graph of g(x) will intersect the y-axis at point (0,4). Being aware of the y-intercept helps in quickly plotting the starting point of the line on the graph, providing a foundation for drawing the rest of the line using the slope.
For function g(x), the y-intercept is 4, as denoted by the equation -x + 4. This means the graph of g(x) will intersect the y-axis at point (0,4). Being aware of the y-intercept helps in quickly plotting the starting point of the line on the graph, providing a foundation for drawing the rest of the line using the slope.
Coordinate Axes
The coordinate axes are a vital part of graphing functions; they consist of two number lines that intersect at right angles. The horizontal line is known as the x-axis, and the vertical one is the y-axis. They meet at the origin, which is the point (0,0).
When graphing functions, we use these axes to plot points where each ordered pair represents a possible solution of the function in the form of (x, y). For instance, for the function f(x) = \( \frac{1}{3}x \), every point on its line represents an x and f(x) (or y) value that makes the function true. The coordinate system allows us to visually interpret the functions and their addition.
When graphing functions, we use these axes to plot points where each ordered pair represents a possible solution of the function in the form of (x, y). For instance, for the function f(x) = \( \frac{1}{3}x \), every point on its line represents an x and f(x) (or y) value that makes the function true. The coordinate system allows us to visually interpret the functions and their addition.
Other exercises in this chapter
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