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Chapter 1: Problem 15

Find the natural domain and graph the functions. $$f(x)=5-2 x$$

Short Answer

Expert verified
The domain is all real numbers. The graph is a line with y-intercept 5 and slope -2.

Step by step solution

01

Identify the function type

The given function is a linear function of the form \(f(x) = ax + b\), where \(a = -2\) and \(b = 5\). Linear functions are defined for all real numbers.
02

Determine the natural domain

The natural domain of linear functions is all real numbers, \(x \in \mathbb{R}\), because there are no restrictions like square roots or divisions by zero in the function.
03

Graph the function

To graph \(f(x) = 5 - 2x\), first identify the y-intercept, \(b = 5\), where the graph touches the y-axis. The slope, \(-2\), indicates that for every one unit increase in \(x\), \(f(x)\) decreases by two units. Plot the point (0,5) and use the slope to find another point, for instance, from (0,5) to (1,3). Draw a line through these points.

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Key Concepts

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

Understanding Linear Functions
Linear functions are the simplest type of functions you'll come across in mathematics. They are characterized by their constant rate of change, which is represented by the slope. A linear function can be written in the form \(f(x) = ax + b\). Here, \(a\) is the slope, and \(b\) is the y-intercept. Let's break these down to understand better:
  • The **slope** \(a\) tells us how steep the line is. If \(a\) is positive, the line goes upwards as you move right. If negative, like in our function, it slopes downwards.
  • The **y-intercept** \(b\) is the point where the line crosses the y-axis. This is where the function value, or \(f(x)\), equals \(b\) when \(x = 0\).
Linear functions have a natural domain of all real numbers, \(x \in \mathbb{R}\), which means they can take any real value as input. This is because there aren't any conditions like denominators or square roots that could restrict the possible values of \(x\).
Graphing Functions Made Simple
Graphing a function means drawing its curve on a graph. For linear functions, this is straightforward as it results in a straight line. Here's how you can do it:
  • **Identify the y-intercept**: This is your starting point. For the function \(f(x) = 5 - 2x\), the y-intercept is \(5\). Mark the point \((0, 5)\) on the y-axis.
  • **Use the slope**: The slope of \(-2\) tells us how to move from the y-intercept. From the point \((0, 5)\), move right 1 unit (positive x-direction) and down 2 units (negative y-direction), getting to point \((1, 3)\).
  • **Draw the line**: Connect these points with a straight line that extends in both directions. This line is the graph of your function.
Remember, the line represents all solutions of the function, as each point on it corresponds to an input-output pair \((x, f(x))\). By graphing, you visualize how the function behaves across its domain.
Slope-Intercept Form Advantage
The slope-intercept form is a powerful tool for working with linear functions. Written as \(y = mx + c\), it clearly highlights the slope \(m\) and the y-intercept \(c\). In our function, \(y = 5 - 2x\) shows:
  • **Slope \(m = -2\)**: This tells us the rate at which \(f(x)\) decreases as \(x\) increases.
  • **Y-intercept \(c = 5\)**: This is where the line crosses the y-axis, or the value of \(f(x)\) when \(x = 0\).
Using this form makes it easy to quickly sketch the graph of a line and understand its behavior. This straightforward approach helps simplify solving problems related to linear functions and ensures a quick way to visualize and analyze the relationships between variables in a linear manner.

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