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Chapter 0: Problem 14

Determine the intercepts of the graphs of the following equations. $$ f(x)=14 $$

Short Answer

Expert verified
y-intercept at (0,14). No x-intercept.

Step by step solution

01

- Understand the Equation

The equation given is a constant function, written as \(f(x) = 14\). This means that the value of \(f(x)\) is always 14, regardless of the value of \(x\).
02

- Find the y-Intercept

The y-intercept occurs where the graph intersects the y-axis. This happens when \(x = 0\). Substituting \(x = 0\) into the equation, we get \(f(0) = 14\). Thus, the y-intercept is at the point \((0, 14)\).
03

- Determine the x-Intercept

The x-intercept occurs where the graph intersects the x-axis. This happens when \(f(x) = 0\). However, since \(f(x) = 14\) and 14 is never equal to 0, there is no x-intercept for this function.

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Key Concepts

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

Y-Intercept
To determine the y-intercept of a function, we look at where it crosses the y-axis. This point occurs when the value of x is zero.
For the given constant function, represented by the equation \( f(x) = 14 \), we substitute x with 0: \( f(0) = 14 \) . This tells us that the function intersects the y-axis at the point (0, 14). In simpler terms, wherever the value of x is 0, the function's value will always be 14. This is the y-intercept.
X-Intercept
The x-intercept is the point where the graph intersects the x-axis. This occurs when the value of the function (y) is zero.
To find the x-intercept for the function \( f(x) = 14 \), we set \( f(x) = 0 \): \( 14 = 0 \) .
This equation is impossible, as 14 cannot equal 0. Therefore, a constant function such as \( f(x) = 14 \) does not have an x-intercept. The graph will not touch or cross the x-axis.
Constant Function
A constant function is a type of function where the output value remains constant, no matter what the input value is. In mathematical terms, it is written as: \( f(x) = c \) where c is a constant.
In the given example, \( f(x) = 14 \), the output value is always 14. Unlike other functions, a constant function does not change its value as x changes. This results in a horizontal line on the graph.
Key characteristics of a constant function:
  • It always produces the same output value.
  • Its graph is a horizontal line.
  • It has a y-intercept but no x-intercept.
Understanding constant functions can help simplify more complex math concepts and improve problem-solving skills.

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