Problem 51 - Elementary and Intermediate Algebra

Question

Find \(g(2)\) and \(g(3) .\) See Example 4. $$ g(x)=x^{2}-10 $$

Step-by-Step Solution

Answer

\( g(2) = -6 \) and \( g(3) = -1 \).

Step 1: Understanding the Problem

The function given is \( g(x) = x^2 - 10 \). We need to find the values of \( g(2) \) and \( g(3) \), which means plugging in \( x = 2 \) and \( x = 3 \) into the function and solving it.

Step 2: Plug x = 2 into the Function

To find \( g(2) \), substitute \( 2 \) into the function \( g(x) = x^2 - 10 \). This gives us:\[ g(2) = 2^2 - 10 \].Calculate \( 2^2 \) which equals \( 4 \). Then subtract \( 10 \) from \( 4 \).

Step 3: Calculate g(2)

Continuing from the previous step, you have:\[ g(2) = 4 - 10 \].Solve this to get:\[ g(2) = -6 \].

Step 4: Plug x = 3 into the Function

To find \( g(3) \), substitute \( 3 \) into the function \( g(x) = x^2 - 10 \). This gives us:\[ g(3) = 3^2 - 10 \].Calculate \( 3^2 \) which equals \( 9 \). Then subtract \( 10 \) from \( 9 \).

Step 5: Calculate g(3)

Continuing from the previous step, you have:\[ g(3) = 9 - 10 \].Solve this to get:\[ g(3) = -1 \].

Key Concepts

Function Evaluation

Function evaluation refers to the process of calculating the output value of a function for a given input value. To evaluate a function, you substitute a specific number for the variable in the function's formula. For instance, with the function given as \[ g(x) = x^2 - 10 \], the task was to find \( g(2) \) and \( g(3) \).
Here's how function evaluation works:

• Identify the given function and the input value you need to evaluate.
• Substitute the given input value into the function wherever you see the variable.
• Perform the arithmetic operations to solve for the function's output.

By following these straightforward steps, you can find the corresponding output for any function input, helping you solve various mathematical problems efficiently.

Substitution

Substitution is a basic yet crucial concept in algebra that involves replacing a variable with a specific number or another expression. When dealing with functions, substitution helps you evaluate the function at different points. In our example, to find \( g(2) \), we replace \( x \) with \( 2 \) in the expression \( x^2 - 10 \).
Let's break down substitution:

• Identify where you need to insert the new value. For this exercise, it was into the function \( g(x) = x^2 - 10 \).
• Carefully replace the variable with the new value, while ensuring to respect any arithmetic rules.
• Execute the calculation to find your answer.

Substitution is not only essential for functions but also plays a key role in solving equations and simplifying expressions in algebra.

Basic Algebra Concepts

Understanding basic algebra concepts helps to build a strong foundation for solving more complex problems. With quadratic functions like \( g(x) = x^2 - 10 \), some fundamental concepts come into play.
Here are a few critical algebra principles used in this exercise:

• Squaring a number: This involves multiplying the number by itself, like \( 2^2 = 4 \) or \( 3^2 = 9 \).
• Arithmetic operations: Once you've substituted and squared the number, basic addition and subtraction follow to simplify the function to its output. For example: when subtracting \( 4 - 10 \) to get \(-6\).
• Function notation: Understanding \( g(x) \) indicates that \( g \) is a function of \( x \), and substituting \( x \) modifies the output of the function.

These basic algebra concepts are pivotal as they provide the skills needed to solve functions, setting a foundation for more advanced mathematical operations.