/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 93 If \(f(x)=x^{2}-2 x+3,\) find ea... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 93

If \(f(x)=x^{2}-2 x+3,\) find each value. $$f(0.5)$$

Short Answer

Expert verified
The value of the function \(f(x)\) at \(x = 0.5\) is \(2.25\).

Step by step solution

01

Substitute the Given Value Into the Function Formula

In the function \(f(x)=x^{2}-2x+3\), replace every \(x\) with the value \(0.5\). This yields: \(f(0.5)=(0.5)^{2}-2(0.5)+3\).
02

Calculate the Value

Now calculate each part of the equation: \((0.5)^{2}=0.25\), \(-2(0.5)=-1\), and adding the constant value \(3\) without any modification. Next, sum up these values: \(0.25 - 1 + 3 = 2.25\). So, \(f(0.5) = 2.25\).

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Key Concepts

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

Understanding Function Evaluation
Function evaluation is like putting a number into a machine and seeing what comes out. When we have a function like \(f(x) = x^2 - 2x + 3\), we're defining a rule that connects each input (the \(x\) value) to exactly one output (the \(f(x)\) value). To evaluate a function, you simply substitute the given number in place of \(x\) and calculate the result. For example, if you're asked to find \(f(0.5)\), you replace each \(x\) in the function with \(0.5\). Remember:
  • The \(x\) you've substituted changes only for this specific evaluation. Your function remains \(f(x) = x^2 - 2x + 3\) for any other input.
  • Function evaluation allows us to see how the function behaves for specific values of \(x\).
The Substitution Method in Functions
The substitution method is essential in mathematics, especially when dealing with functions. It involves replacing a variable in an expression with a given value. Let's break it down: Say you have \(f(x) = x^2 - 2x + 3\) and want to find \(f(0.5)\). The substitution method helps by placing \(0.5\) in each spot where \(x\) appears. This process transforms your initial function into:
  • Start with \(f(0.5) = (0.5)^2\)
  • Next, replace \(-2x\) with \(-2(0.5)\)
  • Finally, add the constant term +3
After substituting, perform the arithmetic to evaluate the expression. Substitution helps you focus on one value, making it easier to handle complex functions step by step.
Solving Algebra Problems: A Methodical Approach
Solving algebra problems often requires a systematic approach. Consider the quadratic function \(f(x) = x^2 - 2x + 3\) and the process of finding \(f(0.5)\). Here's a step-by-step breakdown:
  • First, follow the substitution method to replace \(x\) with \(0.5\).
  • Compute \((0.5)^2 = 0.25\).
  • Next, calculate \(-2(0.5) = -1\).
  • Add the constant 3, so the problem becomes adding these together: \(0.25 - 1 + 3\).
  • Finally, add these results to find the solution: \(0.25 - 1 + 3 = 2.25\).
This approach emphasizes doing one calculation at a time, which reduces the chance of error and makes solving more complex expressions manageable. The steps are straightforward, ensuring you understand each part before moving on to the next, helping simplify what initially may appear complex.

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