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Chapter 2: Problem 52

Find \(g(2)\) and \(g(3)\). \(g(x)=x^{2}-2\)

Short Answer

Expert verified
\( g(2) = 2 \) and \( g(3) = 7 \).

Step by step solution

01

Understand the Function

The given function is \( g(x) = x^2 - 2 \). This is a quadratic function in the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 0 \), and \( c = -2 \). Our task is to find the values of \( g(2) \) and \( g(3) \).
02

Calculate \( g(2) \)

To find \( g(2) \), substitute \( x = 2 \) into the function. This gives us: \[ g(2) = (2)^2 - 2 = 4 - 2 = 2 \] Thus, \( g(2) = 2 \).
03

Calculate \( g(3) \)

Next, substitute \( x = 3 \) into the function to find \( g(3) \). This results in: \[ g(3) = (3)^2 - 2 = 9 - 2 = 7 \] Therefore, \( g(3) = 7 \).

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Key Concepts

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

Function Evaluation
Function evaluation is the process of determining the output of a function for a specific input value. In this context, we deal with a quadratic function defined as \( g(x) = x^2 - 2 \).
To evaluate this function, you select particular input values for \( x \) and compute their corresponding outputs, which are \( g(2) \) and \( g(3) \).
Function evaluation is particularly useful in mathematics because it allows you to understand how a particular input affects the function's outcome.
  • Choose the values of \( x \) you wish to evaluate, such as 2 and 3 in this exercise.
  • Replace \( x \) in the function with these values to find the corresponding outputs.
This simple process lets you explore the behavior of functions and predict future outputs.
Substitution in Algebra
In algebra, substitution is a method used to simplify expressions or solve equations by replacing variables with their actual values.
When working with functions like \( g(x) = x^2 - 2 \), substitution involves plugging in the chosen values of \( x \) into the function to simplify and evaluate it.
Substitution can be broken down into easy steps:
  • Identify the variable you need to substitute in, which is \( x \) in our function.
  • Replace every occurrence of this variable in the function with the given number, such as 2 or 3.
  • Perform the arithmetic operations to simplify the expression and find the result.
By carefully substituting values into a function, you can solve problems systematically and confidently.
Understanding substitution also builds a strong foundation for more complex algebraic manipulations.
Value of a Function
The value of a function refers to the output you receive after evaluating a function at a specific input. It's essentially the answer you get after performing function evaluation and substitution.
For example, for the function \( g(x) = x^2 - 2 \), when you find \( g(2) \), the calculated value is 2.
Similarly, the computation of \( g(3) \) gives you a function value of 7.
It's important to underline that the value of a function solely depends on the input values substituted into the expression.
  • Every input \( x \) will yield a unique output based on the function's rule.
  • These values can help in graphing the function or understanding its characteristics such as roots, intercepts, and vertex in the case of quadratic functions.
This understanding is crucial as it allows you to predict and interpret how functions behave, which is essential in both pure and applied mathematics.

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