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Chapter 11: Problem 32

The first coordinate of an x-intercept of a function is a(n) ________ of the function.

Short Answer

Expert verified
Zero or root

Step by step solution

01

Understand X-Intercept

An x-intercept is a point where the graph of a function crosses the x-axis. This means at the x-intercept, the y-coordinate is 0.
02

Set Y to Zero

To find the x-intercept, set the function equal to 0 and solve for x. Mathematically, if the function is f(x), then set f(x) = 0 and solve for x.
03

Define the Term

The value of x obtained when setting the function to 0 is a root or zero of the function. This is because it is the point where the function changes sign.

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Key Concepts

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

roots of a function
The 'roots of a function' are the values of x that make the function equal to zero. These are the points where the graph of the function intersects the x-axis. The roots are fundamental to understanding the behavior of a function.
Roots are also called 'solutions' or 'x-intercepts'.
For example, if we have a function \(f(x) = x^2 - 4\), the roots can be found by setting the equation equal to zero: \(x^2 - 4 = 0\). Solving this gives us the values x = 2 and x = -2. Hence, the roots of the function are 2 and -2.
Understanding the roots helps to determine important aspects such as:
  • The points where the graph touches or crosses the x-axis.
  • Intervals where the function is positive or negative.
  • Behavior changes in the function.
    • zero of a function
    The 'zero of a function' is another way to refer to the roots. When we say the zero of a function, we are looking for the point(s) where the function equals zero.
    This point is significant because it indicates where the function's value is zero, meaning no value (zero) is present.
    For instance, with the function \(f(x) = 2x - 6\), we find the zero by setting the function equal to zero: \(2x - 6 = 0\). Solving this equation gives us x = 3. Thus, the zero of the function is 3.
    Knowing the zeroes helps to:
  • Understand the solution to algebraic equations.
  • Predict the behavior of the function.
    • solving for x
    When we talk about 'solving for x,' we are finding the value of x that makes the function equal to a certain value, typically zero.
    This is a crucial step to identify the roots or zeroes of a function.
    For example, solving the equation \(x^2 - 9 = 0\) involves rearranging and simplifying the expression:
  • Step 1: Add 9 to both sides to get \(x^2 = 9\).
  • Step 2: Take the square root of both sides to get \(x = 3\) and \(x = -3\).

    • Therefore, x = 3 and x = -3 are the solutions.
    Solving for x is essential in many areas of mathematics and real-world problem-solving:
  • Finding where a function intersects the x-axis.
  • Identifying maximum and minimum points in calculus.
  • Using algebraic methods to model and solve real-life problems.

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