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Chapter 4: Problem 27

Determine the zeros, if any, of each function. $$ f(x)=2.7-x $$

Short Answer

Expert verified
The zero of the function is x = 2.7.

Step by step solution

01

Understand the Problem

The goal is to determine the zeros of the function. A zero of a function is a value of x that makes the function equal to zero.
02

Set the Function Equal to Zero

Set the given function equal to zero: \[ f(x) = 2.7 - x = 0 \]
03

Solve for x

Solve the equation from the previous step for x: \[ 2.7 - x = 0 \]Add x to both sides to isolate the variable:\[ 2.7 = x \]
04

Verify the Solution

Substitute x = 2.7 back into the original function to verify: \[ f(2.7) = 2.7 - 2.7 = 0 \]Since the function equals zero, x = 2.7 is a correct solution.

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Key Concepts

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

solving linear equations
Linear equations are the simplest type of algebraic equations. They can be written in the form \( ax + b = 0 \), where \( a \) and \( b \) are constants. The objective is to find the value of \( x \) that makes the equation true. Solving linear equations often involves isolating the variable by performing arithmetic operations. This can involve adding, subtracting, multiplying, or dividing both sides of the equation. For example, in the equation \( 2.7 - x = 0 \), we can solve for \( x \) by adding \( x \) to both sides:

\[ 2.7 - x + x = 0 + x \],

simplifying to \[ 2.7 = x \].
By following these steps, we can systematically solve linear equations.
function zeros
The zero of a function, also called the root, is a value of \( x \) that makes the function equal to zero. In other words, it's where the graph of the function touches or crosses the x-axis. To find the zero of a function, set the function equal to zero and solve for \( x \). For example, to find the zero of the function \( f(x) = 2.7 - x \), we set

\[ f(x) = 0 \],

\[ 2.7 - x = 0 \],

and solve for \( x \):

\[ 2.7 - x = 0 \], \[ 2.7 = x \].
The zero of this function is \( x = 2.7 \). This means when \( x = 2.7 \), the function is equal to zero. This point on the graph is where the function crosses the x-axis.
algebraic verification
Algebraic verification is the process of checking that your solution is correct by substituting the value back into the original equation. It ensures that no mistakes were made during the solving process. For our function \( f(x) = 2.7 - x \), we found that the zero is \( x = 2.7 \). So, we substitute \( x = 2.7 \) back into the function to verify:

\[ f(2.7) = 2.7 - 2.7 = 0 \].
Since the function equals zero at \( x = 2.7 \), this confirms that our solution is correct. Verification is a crucial step in solving any mathematical problem because it validates our results and helps us catch any mistakes we might have made along the way.

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Most popular questions from this chapter

Review exponential notation and order of operations (Section \(1.8)\) Simplify. [ 1.8] $$ -2^{6} $$

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To prepare for Section \(4.3,\) review simplifying algebraic expressions (Section \(1.8)\) Simplify. [ 1.8] $$ 5(2 x+3 y)-3(7 x+5 y) $$
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