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Chapter 1: Problem 19

Sketch the graph of \(f\) by hand. $$f(x)=3$$

Short Answer

Expert verified
The graph of \(f(x) = 3\) is a horizontal line at \(y = 3\).

Step by step solution

01

Identify the Function Type

The given function is a constant function because it does not have any variable terms, and is expressed in the form \(f(x) = c\) where \(c\) is a constant. In this case, \(f(x) = 3\).
02

Understand the Graph of a Constant Function

A constant function like \(f(x) = 3\) represents a horizontal line on the graph. The function value is the same for all \(x\)-values. This means that regardless of the \(x\) value, the output \(f(x)\) will always be 3.
03

Determine the Y-intercept

For the function \(f(x) = 3\), the graph touches the y-axis at \((0, 3)\). This point is called the y-intercept and represents where the graph will cross the y-axis.
04

Draw the Horizontal Line

On a graph, mark the point \((0, 3)\) on the y-axis, since this is the y-intercept. Then, draw a horizontal line across the plane that passes through the y-value of 3. This line should extend infinitely to the left and right, parallel to the x-axis.

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Key Concepts

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

graphing functions
Graphing functions is a foundational concept in mathematics that helps us visualize how functions behave. When graphing functions, we represent them as lines or curves on a coordinate plane. For each input value, or 'x' value, we have a corresponding output value or 'y' value, which we plot as a point.
Graphing makes it easy to see patterns or trends. For example, with a linear function, you'll see a straight line. When considering a function like the constant function \(f(x) = 3\), its graph focuses on giving a clearer visual of constant values and behaviors.
  • Start by plotting points on a coordinate plane.
  • Ensure there's a consistent relationship between x and y values.
  • Connect the points smoothly if applicable (though, for constant functions, you'll see a straight line).

Visualizing the function helps to identify key characteristics such as intercepts and the overall shape.
y-intercept
A y-intercept is a crucial point where the graph of a function crosses the y-axis. In mathematics, this is the \((0, y)\) coordinate. The y-intercept occurs when the input value \(x\) is zero. This means you look at where the function intersects the vertical y-axis.
For constant functions, like \(f(x) = 3\), identifying the y-intercept is straightforward. Here, the y-intercept is the point \((0, 3)\), representing the function's output when the input is zero.
  • Examine the function equation and set \(x = 0\) to find the y-intercept.
  • Plot this point on the y-axis.
  • The y-coordinate at this point gives the function's constant value.

Being able to find and plot the y-intercept aids in understanding the position and height of the function on the graph.
horizontal line
A horizontal line is a straight line that runs parallel to the x-axis. This kind of line shows that a function maintains a constant value, regardless of the x-values. It's a key feature of constant functions. For \(f(x) = 3\), the graph is a horizontal line positioned at y = 3.
Horizontal lines reflect great stability in function values, which means the output does not change across different x-values.
  • Locate the function's y-value.
  • Draw a line parallel to the x-axis at this y-value.
  • This line extends infinitely in both horizontal directions.

Understanding horizontal lines clarifies how constant functions behave and appear graphically. These lines emphasize that while x changes, y remains steady.

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Most popular questions from this chapter

Find \(f(a), f(b+1),\) and \(f(3 x)\) for the given \(f(x)\) $$f(x)=x^{2}$$

Using interval notation, write each set. Then graph it on a number line. $$\\{x | 1 \leq x<2\\}$$

Asian-American populations (in millions) are shown in the table. $$\begin{array}{|l|c|c|c|c|}\hline \text { Year } & 2003 & 2005 & 2007 & 2009 \\\\\hline \begin{array}{l}\text { Population } \\\\\text { (in millions) }\end{array} & 11.8 & 12.6 & 13.3 & 14.0\end{array}$$ (a) Use the points \((2003,11.8)\) and \((2009,14.0)\) to find the point-slope form of a line that models the data. \(\operatorname{Let}\left(x_{1}, y_{1}\right)=(2003,11.8)\) (b) Use this equation to estimate the Asian-American population in 2013 to the nearest tenth of a million.

Using the variable \(x\), write each interval using set-builder notation. $$(-4,3)$$

Using the variable \(x\), write each interval using set-builder notation. $$(3, \infty)$$
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