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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
The y-intercept is (0, 14). There are no x-intercepts.

Step by step solution

01

Understand the Equation

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

Find the y-intercept

To find the y-intercept, set x to 0. We have: \(f(0) = 14\). Therefore, the y-intercept is the point (0, 14).
03

Find the x-intercepts

To find the x-intercepts, set f(x) to 0 and solve for x: \(14 = 0\). This equation is not possible, hence there are no x-intercepts.

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Key Concepts

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

y-intercept
The y-intercept of a function is the point where the graph of the function crosses the y-axis. This happens when the value of x is zero.
For any function, to find the y-intercept, we can simply substitute x with 0 and solve for the value of the function.

In our given equation, which is a constant function, we have:
$$f(x) = 14$$

By substituting x with 0, we get: $$f(0) = 14$$ This tells us that the y-intercept is at the point (0, 14).
This means, no matter what the value of x is, as soon as x is 0, the function will always cross the y-axis at a height of 14.
This forms a horizontal line at y = 14, which helps us visualize the constant nature of the function.
x-intercept
The x-intercept of a function is the point where the graph of the function crosses the x-axis. This happens when the value of the function ($$f(x)$$) equals zero.
To find the x-intercept, we set the function equal to 0 and solve for x.

For our given constant function, $$f(x) = 14$$
we set it to zero:
$$14 = 0$$
This equation is impossible, as 14 can never equal 0. Thus, no x-intercepts exist for constant functions where the function's value is not zero.

This means that the graph does not touch the x-axis at any point, emphasizing the flat, unchanging nature of the constant function in this case.
constant function
A constant function is one where the output value remains the same for all input values.
In mathematical terms, it is written as $$f(x) = c$$, where $$c$$ is a constant.
This results in a horizontal line on the graph, as the y-value (output) does not change regardless of the x-value (input).
In our example, $$f(x) = 14$$, the output is always 14. This means:
  • The graph is a horizontal line at y = 14.
  • The y-intercept is always at the point (0, 14).
  • There are no x-intercepts since the line does not cross the x-axis.

This simplicity makes constant functions easy to work with, and understanding their intercepts is straightforward.
Their consistent value is a powerful tool in many mathematical scenarios.

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