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Chapter 16: Problem 9

Evaluate the function for the given value of \(x\). $$f(x)=x^{2}-2 x+1 ; x=3$$

Short Answer

Expert verified
The function \(f(x) = x^{2} - 2x + 1\) evaluated at \(x = 3\) equals 4.

Step by step solution

01

Identification of the Function and Variable

Identify the given function, which in our case is \(f(x) = x^{2} - 2x + 1\), and the given value of the variable, which is \(x=3\).
02

Substitution

Substitute the given value \(x=3\) into the function \(f(x)\). Which results in: \(f(3) = (3)^{2} - 2*(3) + 1\)
03

Calculation

Now proceed with simple arithmetic operations: \(f(3) = 9 - 6 +1\)
04

Simplify Further

By performing these operations, obtain the final result: \(f(3) = 4\)

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Key Concepts

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

Function Evaluation
Understanding how to evaluate functions is a fundamental skill in algebra. Function evaluation is the process of determining the output of a function for a specific value of its variable, or input. In the context of our exercise, the function given is f(x) = x^2 - 2x + 1, and the objective is to find out the value of this function when x is equal to 3. This process requires substituting the given input value into the function and then calculating using the appropriate arithmetic operations to find the output, which is also known as the value of the function at x = 3. In the solution, we saw that f(3) = 4, which is the result of evaluating the function at that particular point.
Substitution Method
The substitution method is a vital tool when working with functions. This technique involves replacing the variable in a function with a given number to perform an operation. In our exercise, we used the substitution method by replacing every occurrence of x in the function with the number 3. It is crucial to substitute accurately to ensure the function is evaluated correctly. After substitution, the original expression x^2 - 2x + 1 becomes 3^2 - 2(3) + 1. Once the substitution is done, the next steps involve arithmetic operations to simplify the expression to reach the final answer.
Arithmetic Operations
The core of evaluating functions often lies in performing arithmetic operations. Once the substitution is made, we must carry out the operations in the correct order. Remembering the order of operations鈥攑arentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS)鈥攊s essential. In our exercise, after substituting x = 3, we calculate the square of 3, perform the multiplication, and then proceed with the addition and subtraction. We calculated (3)^2 to get 9, subtracted 6 (which is 2*3), and finally added 1 to get the result 4. These steps follow the systematic approach of arithmetic operations necessary for simplifying the expression.
Simplifying Expressions
Finally, simplifying expressions is all about reducing them to their simplest form. This involves combining like terms and performing arithmetic operations until no further simplification is possible. In the given exercise, after the initial arithmetic operations were completed, we were left with the simplified form f(3) = 4. It鈥檚 important to go step by step, to ensure no errors are made during simplification. With each step carefully executed, the simplification process leads to the accurate evaluation of the function for the given value of its variable.

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