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Chapter 1: Problem 13

If \(f(x)=(2 x-1)^{2},\) evaluate \(f(0), f(1),\) and \(f(-2)\)

Short Answer

Expert verified
f(0) = 1, f(1) = 1, f(-2) = 25.

Step by step solution

01

Evaluate f(0)

First, substitute 0 into the given function. So, calculate \[ f(0) = (2(0) - 1)^2 \]. Solve inside the parentheses first: \[ 2(0) - 1 = -1 \]. Now, square it: \[ (-1)^2 = 1 \]. Therefore, \[ f(0) = 1 \].
02

Evaluate f(1)

Next, substitute 1 into the given function. Calculate \[ f(1) = (2(1) - 1)^2 \]. Solve inside the parentheses first: \[ 2(1) - 1 = 1 \]. Now, square it: \[ (1)^2 = 1 \]. Therefore, \[ f(1) = 1 \].
03

Evaluate f(-2)

Finally, substitute -2 into the given function. Calculate \[ f(-2) = (2(-2) - 1)^2 \]. Solve inside the parentheses first: \[ 2(-2) - 1 = -5 \]. Now, square it: \[ (-5)^2 = 25 \]. Therefore, \[ f(-2) = 25 \].

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Key Concepts

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

Function Evaluation
Evaluating functions is a fundamental concept in algebra. When we evaluate a function, we determine the function's value for a given input. Let's take the function from the original exercise: \( f(x) = (2x - 1)^2 \). To evaluate \( f(0) \), \( f(1) \), and \( f(-2) \), follow these simple steps:
Function Evaluation Step-by-Step
1. Substitute the input value into the function.
2. Simplify the expression to find the output.

For example, to evaluate \( f(0) \):
    First, substitute 0 into the function: \( f(0) = (2(0) - 1)^2 \).
    Next, simplify the expression inside the parentheses: \( 2(0) - 1 = -1 \).
    Finally, square the result: \( (-1)^2 = 1 \).
    So, \( f(0) = 1 \).
Repeat these steps for \( f(1) \) and \( f(-2) \) as shown in the original solution.
Quadratic Functions
A quadratic function is a type of polynomial function that has the form \( ax^2 + bx + c \). In our example, \( f(x) = (2x - 1)^2 \) is a special kind of quadratic function. When expanded, it follows the structure of a quadratic equation:

  • Quadratic functions can create parabolic graphs.
  • They always have a highest or lowest point known as the vertex.
  • They have an axis of symmetry that passes through the vertex.

Understanding the properties of quadratic functions helps in evaluating them correctly.
Substitution
Substitution is the process of replacing a variable in an expression with a given value. This is the key step in evaluating functions. To substitute correctly, follow these guidelines:
  • Identify the variable in the function.
  • Replace the variable with the given number.
  • Simplify the expression step by step.

For example, to evaluate \( f(-2) \), substitute \(-2\) for \( x \) in the function \( f(x) = (2x - 1)^2 \):
  • First, replace \( x \) with \( -2 \): \( f(-2) = (2(-2) -1)^2 \).
  • Next, simplify inside the parentheses: \( 2(-2) - 1 = -5 \).
  • Finally, square the result: \( (-5)^2 = 25 \).
Therefore, \( f(-2) = 25 \). This method applies to any function substitution.

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