Problem 7 - Algebra for College Students

Question

Complete each ordered pair so that it satisfies the given equation. $$f(x)=x^{2} \quad(4, \quad),(\quad, 9)$$

Step-by-Step Solution

Answer

The complete ordered pairs are \((4, 16)\) and either \((3, 9)\) or \((-3, 9)\).

Step 1: Understand the given equation

The given function is defined as follows: \[ f(x) = x^2 \] This equation tells us that for any input value \( x \), the output \( f(x) \) is the square of \( x \).

Step 2: Complete the first ordered pair

The first ordered pair is \((4, \quad)\). To find the missing second component, substitute \( x = 4 \) into the function: \[ f(4) = 4^2 = 16 \]Therefore, the first ordered pair is \( (4, 16) \).

Step 3: Complete the second ordered pair

The second ordered pair is \((\quad, 9)\). To find the missing first component, set \( f(x) = 9 \) and solve for \( x \): \[ x^2 = 9 \] This gives two possible solutions for \( x \): \[ x = 3 \] or \[ x = -3 \]Therefore, the second ordered pair can be either \( (3, 9) \) or \( (-3, 9) \).

Key Concepts

Function Notation

Function notation is a way to describe the relationship between inputs and outputs in a function.
It looks like this: \[ f(x) \]
Where:

• \( f \) represents the function.
• \( x \) is the input variable, also known as the independent variable.

The entire expression \( f(x) \) gives us the output, also called the dependent variable. It's essential to understand that the function tells us what to do with the input. For example, \( f(x) = x^2 \) shows that we should square the input variable \( x \) to get the output.
This makes it easy to calculate the results for different inputs. If \( x = 4 \), then \( f(4) = 4^2 = 16 \). Similarly, if \( f(x) = 9 \), then the input \( x \) must be either \( 3 \) or \( -3 \).

Function notation helps us to keep things simple and organized, especially when dealing with more complex mathematical problems.
Learning how to read and use function notation properly will make solving equations much more intuitive.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation.
For example, if we have an equation like \( f(x) = x^2 \), and we know that \( f(x) = 9 \), we need to find the value of \( x \) that makes this statement true.
This process generally involves a few steps:

• Identify what you're solving for.
• Isolate the variable on one side of the equation.
• Perform the necessary mathematical operations to solve for the variable.

In our example, if \( f(x) = 9 \), we need to solve the equation \( x^2 = 9 \). By taking the square root of both sides, we get: \[ x = 3 \] or \[ x = -3 \].

So, the solutions for \( x \) are either \( 3 \) or \( -3 \). In the context of functions, solutions help to form ordered pairs. For instance, if \( f(x) = 9 \), our ordered pairs would be \( (3, 9) \) and \( (-3, 9) \). This approach applies to a wide range of mathematical problems beyond just functions.

Squaring Numbers

Squaring a number means multiplying the number by itself.
This operation is commonly written using an exponent of 2, such as \( x^2 \).
For example, if \( x = 4 \), then \( x^2 = 4 \times 4 = 16 \).

Let's break it down:

• Start with your number.
• Multiply the number by itself.
• The result is the square of that number.

Squaring is a straightforward process but plays a crucial role in many mathematical concepts, including functions and equations.
In the function \( f(x) = x^2 \), squaring is the operation we perform on the input \( x \) to get the output \( f(x) \).

Remember that squaring a positive or negative number will always give a non-negative result, since \( (-3)^2 = 9 \) and \( 3^2 = 9 \). This notion is vital when solving equations like \( x^2 = 9 \), where the solutions are both \( 3 \) and \( -3 \).
Mastering squaring numbers will make it easier to understand a wide range of topics in mathematics.