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Chapter 4: Problem 4

Given the function defined by the rule \(\mathrm{f}(\mathrm{x})=-2\), evaluate \(\mathrm{f}(0), \mathrm{f}(\mathrm{b})\), and \(\mathrm{f}(5-4 \mathrm{x})\), then draw the graph of \(\mathrm{f}\).

Short Answer

Expert verified
The function outputs are all -2, and the graph is a horizontal line at \(y = -2\).

Step by step solution

01

Evaluate f(0)

Since the function is defined by the rule \(f(x) = -2\), this means that for any input \(x\), the output will always be \(-2\). Thus, \(f(0) = -2\).
02

Evaluate f(b)

Apply the same rule to \(f(b)\). Regardless of the value of \(b\), the function yields \(-2\) as output, so \(f(b) = -2\).
03

Evaluate f(5-4x)

Similarly, use the function rule on \(f(5-4x)\). The output for any expression of \(5-4x\) is also \(-2\). Therefore, \(f(5-4x) = -2\).
04

Draw the Graph of f

Since the function \(f(x) = -2\) is constant, the graph is a horizontal line. Plot several points (e.g., \((0, -2), (1, -2), (-1, -2)\)), and draw a line through these points horizontally at \(y = -2\) on the coordinate plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
Function evaluation is the process of finding the output of a function for specific inputs. In this exercise, the function is defined as \(f(x) = -2\). This indicates that irrespective of the input value given to \(x\), the outcome will always be \(-2\). You can see this principle applied in evaluating different expressions. For example:
  • When \(x = 0\), the function evaluates as \(f(0) = -2\).
  • When \(x = b\) (where \(b\) is any real number), the output remains \(f(b) = -2\).
  • Similarly, when \(x = 5 - 4x\), the function still produces \(f(5-4x) = -2\).
This demonstrates that a constant function returns the same output for any input, reflecting its unchanging nature. Understanding this provides the basic groundwork for evaluating constant functions consistently.
Graphing Functions
Graphing functions involve drawing the function's output on a coordinate plane, showcasing the relationship between input and output values visually. In the case of \(f(x) = -2\), the graphing process is simplified due to its constant nature. Here is how you can do it:
  • First, recognize that the function produces the same output, \(-2\), regardless of the input.
  • To graph \(f(x) = -2\), plot several points on the graph: for instance,
    • When \(x = 0\), \(f(x) = -2\), plot the point \( (0, -2) \).
    • Choose another value like \(x = 1\), again \(f(x) = -2\), resulting in the point \( (1, -2) \).
    • Try \(x = -1\), and still, the output is \(-2\), giving the point \( (-1, -2) \).
Connect these points, creating a straight line parallel to the x-axis at \(y = -2\). Graphing helps to visually confirm the constant behavior of the function across various inputs.
Horizontal Line
In mathematics, a horizontal line is a straight line that runs parallel to the x-axis. It implies that no matter how the x-value changes, the y-value remains constant. For the function \(f(x) = -2\), which is a constant function, the graph reflects this characteristic.
  • Every point on the line has the same y-coordinate, exemplifying why it's called horizontal.
  • The graph of \(f(x) = -2\) is a horizontal line positioned at \(y = -2\). This means, irrespective of the x-values involved, any point chosen along this line will have a consistent output of \(-2\).
Understanding horizontal lines and how they relate to constant functions is crucial in visualizing how constancy affects graphical representations. It underscores the fact that the variation in x does not influence the y-value in such functions.

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Most popular questions from this chapter

Given the function defined by the rule $$f(x)=\left\\{\begin{array}{ll}0, & \text { if } x<0 \\\2, & \text { if } x \geq 0\end{array}\right.$$ evaluate \(\mathrm{f}(-2), \mathrm{f}(0)\), and \(\mathrm{f}(3)\), then draw the graph of \(\mathrm{f}\) on a sheet of graph paper. State the domain and range of \(\mathrm{f}\).

The speed of a skateboarder as she travels down a slope is a function of time and is described by the constant function \(\mathrm{v}(\mathrm{t})\) \(=8\), where \(t\) is measured in seconds and \(v\) is measured in feet per second. Draw the graph of \(v\) versus \(t\). Be sure to label each axis with the appropriate units. Shade the area under the graph of \(\mathrm{v}\) over the time interval \([0,60]\) seconds. What is the area under the graph of \(v\) over this time interval and what does it represent?

Given the function defined by the rule \(\mathrm{h}(\mathrm{x})=-4\), evaluate \(\mathrm{h}(-2), \mathrm{h}(\mathrm{a})\), and \(\mathrm{h}(2 \mathrm{x}+3)\), then draw the graph of \(\mathrm{h}\).

Given the function defined by the rule \(\mathrm{f}(\mathrm{x})=3\), evaluate \(\mathrm{f}(-3), \mathrm{f}(0)\) and \(\mathrm{f}(4)\), then sketch the graph of \(\mathrm{f}\).

The speed of an automobile traveling on the highway is a function of time and is described by the constant function \(\mathrm{v}(\mathrm{t})=\) 30, where \(t\) is measured in hours and \(\mathrm{v}\) is measured in miles per hour. Draw the graph of \(\mathrm{v}\) versus t. Be sure to label each axis with the appropriate units. Shade the area under the graph of \(\mathrm{v}\) over the time interval \([0,5]\) hours. What is the area under the graph of \(\mathrm{v}\) over this time interval and what does it represent?
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