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Chapter 9: Problem 24

Graph each polynomial function. Give the domain and range. $$ f(x)=3 x+2 $$

Short Answer

Expert verified
The graph is a straight line with domain \((-\infty, \infty)\) and range \((-\infty, \infty)\).

Step by step solution

01

- Identify the Type of Polynomial

Observe the given function: \( f(x) = 3x + 2 \). This is a linear polynomial, as it is in the form \( ax + b \).
02

- Determine the Slope and Intercept

In the function \( f(x) = 3x + 2 \), the coefficient of \( x \) is the slope (3), and the constant term (2) is the y-intercept. Thus, the slope is 3 and the y-intercept is 2.
03

- Plot the Y-Intercept

Start by plotting the y-intercept (0, 2) on the graph. This is the point where the line crosses the y-axis.
04

- Use the Slope to Plot Another Point

From the y-intercept, use the slope to find another point. A slope of 3 means that for every 1 unit increase in \( x \), \( y \) increases by 3 units. Moving 1 unit to the right from (0, 2) reaches (1, 5).
05

- Draw the Line

Draw a straight line through the points (0, 2) and (1, 5). This line represents the function \( f(x) = 3x + 2 \).
06

- State the Domain

For a linear polynomial, the domain consists of all real numbers (\( -\infty, \infty \)).
07

- State the Range

Similarly, since the line extends infinitely in both directions vertically, the range is also all real numbers (\( -\infty, \infty \)).

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Key Concepts

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

Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. In simple terms, it looks like a series of terms where each term is a constant multiplied by the variable raised to a non-negative integer power.
For example:
  • 3x2 + 2x + 1
  • 5x + 7
  • x3 - 4x2 + 3x - 2
Each of these is a polynomial function of a single variable x. The degree of the polynomial is the highest power of the variable in the expression. For instance, the degree of 3x2 + 2x + 1 is 2. Polynomial functions can be classified based on their degree, such as:
  • 0 degree: Constant polynomial (e.g., 7)
  • 1st degree: Linear polynomial (e.g., 3x + 2)
  • 2nd degree: Quadratic polynomial (e.g., x2 - 4x + 4)
  • 3rd degree: Cubic polynomial (e.g., x3 + 2x - 1)
Understanding polynomial functions is essential, as they appear frequently in both pure and applied mathematics.
Linear Polynomial
A linear polynomial is a polynomial of degree 1. This means it has the form f(x) = ax + b, where a and b are constants, and x is the variable. The highest power of x in this polynomial is 1.
For example, in the exercise f(x) = 3x + 2, this function is linear because it fits the form ax + b. Here, a = 3 and b = 2.
Key characteristics of a linear polynomial include:
  • Slope: The coefficient of x, which determines the steepness of the line. In f(x) = 3x + 2, the slope is 3.
  • Y-Intercept: The constant term, which is the point where the line crosses the y-axis. In our function, the y-intercept is 2.
  • Graph: The graph of a linear polynomial is always a straight line.
Graphing a linear polynomial function involves plotting the y-intercept and using the slope to find another point on the line. For instance, starting at (0, 2) and going up 3 units and over 1 unit, you find another point (1, 5). Drawing a line through these points gives the graph of the linear polynomial.
Domain and Range
The domain and range are important concepts when discussing functions. They describe the set of possible input values and output values, respectively.
For a linear polynomial like f(x) = 3x + 2:
  • Domain: The domain of a function is the set of all possible x-values that can be input into the function. Since you can input any real number into a linear polynomial, the domain is all real numbers, \(\(-\infty, \infty\)\).
  • Range: The range is the set of all possible y-values the function can produce. For linear polynomials, as the line extends infinitely in both vertical directions, the range is also all real numbers, \(\(-\infty, \infty\)\).
Understanding the domain and range helps when graphing and analyzing functions, as it gives insight into the behavior of the function over all possible inputs. For example, in graphing the function f(x) = 3x + 2, it clarifies that the line will continue infinitely in both directions, covering all y-values.

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