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Chapter 4: Problem 23

Determine the zeros, if any, of each function. $$ f(x)=x-5 $$

Short Answer

Expert verified
The zero of the function is x = 5.

Step by step solution

01

- Understand the problem

We need to find the zeros of the function. A zero of the function is a value of x that makes the function equal to zero.
02

- Set the function equal to zero

To find the zeros, set the given function equal to zero: f(x) = x - 5 0 = x - 5.
03

- Solve for x

Solve the equation 0 = x - 5 by adding 5 to both sides of the equation: 0 + 5 = x - 5 + 5 5 = x.

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Key Concepts

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

Algebraic Equations
Algebraic equations form the foundation of algebra. They are mathematical statements that assert the equality of two expressions. In the context of finding zeros of a function, we're dealing with equations in one variable, typically represented as \( x \). The given equation in this exercise is \( f(x) = x - 5 \).
Here, \( f(x) \) represents the function, and we are tasked with finding the value of \( x \) that makes this function zero.
  • First, we set the function equal to zero: \( x - 5 = 0 \).
  • This is an algebraic equation where we need to solve for \( x \).
  • Solving it means finding the value of \( x \) satisfying this equation.
Understanding algebraic equations is crucial as they appear in various forms throughout mathematics. They can be linear, quadratic, or even more complex. Starting with simple ones, like in this exercise, helps build a strong foundation.
Solving Linear Equations
Solving linear equations is an essential skill in elementary algebra. A linear equation is one where the highest power of the variable is one. In this exercise, the linear equation we work with is \( x - 5 = 0 \).
The steps to solve it are straightforward:
  • First, we isolate the variable, in this case, \( x \).
  • We do this by performing operations that keep the equation balanced. For \( x - 5 = 0 \), we add 5 to both sides to isolate \( x \).
  • Adding 5 to both sides yields \( x = 5 \).
This process involves understanding inverse operations. Here, subtraction 'inversed' by addition. Practicing these steps with varied equations helps master the skill.
Elementary Algebra
Elementary algebra is the branch of mathematics that deals with simple operations and their properties. It lays the groundwork for more advanced topics.
Key concepts include working with variables, understanding equations, and performing basic operations. In the given exercise, let's review fundamental ideas:
  • Variables represent unknown quantities, commonly denoted by \( x \), \( y \), etc.
  • Algebra involves forming expressions and equations with these variables.
  • Solving equations often requires isolating the variable, as shown in the step-by-step solution.
Practicing these basic operations enriches problem-solving skills. Recognizing patterns helps too. By mastering elementary algebra, students can confidently approach more complex mathematical concepts.

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Most popular questions from this chapter

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Solve each system using the substitution method. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this. $$ \begin{aligned} &2 x+3 y=-2\\\ &2 x-y=9 \end{aligned} $$

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