Problem 11 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 9-14, y=f(x) .\) Find \(f(-3)\) for each function. \(y=1+x^{2}\)

Step-by-Step Solution

Answer

The value of \( f(-3) \) is 10.

Step 1: Understand the Function

First, identify the given function. The function is given as \( y = f(x) = 1 + x^2 \). This means, for any value of \( x \), \( f(x) = 1 + x^2 \).

Step 2: Substitute \( x = -3 \)

To find \( f(-3) \), substitute \( x = -3 \) into the function. This gives us \( f(-3) = 1 + (-3)^2 \).

Step 3: Calculate \((-3)^2\)

Evaluate \((-3)^2\). Since squaring a negative number results in a positive number, \((-3)^2 = 9\).

Step 4: Add to Find Result

Add the square result to 1 as per the function definition. Calculating \( 1 + 9 \) gives \( f(-3) = 10 \).

Key Concepts

Substitution

In mathematics, substitution refers to the process of replacing a variable with a given value. This is a fundamental concept, especially when you need to evaluate functions or solve equations. In this exercise, we are dealing with a function, denoted as \( y = f(x) = 1 + x^2 \). To evaluate this function for a specific value, in this case \( x = -3 \), we substitute -3 in place of \( x \) in the function expression. This means we replace every instance of \( x \) with -3, transforming our expression into \( f(-3) = 1 + (-3)^2 \).

• Why Substitute? - Substitution allows us to compute the specific outputs (or function values) based on particular inputs, providing us with an exact result for the function at that point.
• Example: - If you have a function \( f(x) = 1 + x^2 \) and you want to know what happens when \( x \) is -3, substitution gives us \( 1 + (-3)^2 \).

Always remember, substitution is a straightforward method that transforms our symbolic expression into a numeric value, simplifying the problem-solving process.

Squaring Numbers

Squaring a number means multiplying the number by itself. This is a crucial arithmetic operation, especially when working with quadratic functions like \( y = 1 + x^2 \). Here, our task was to square \( -3 \), expressed as \((-3)^2\).

• Steps to Square: Start by multiplying the number by itself. For -3, this becomes \( (-3) \times (-3) \).
• Result of Squaring: The squaring process changes a negative number into a positive one because the multiplication of two negative numbers results in a positive number. Thus, \( (-3)^2 = 9 \).
• Application: In the function \( f(x) = 1 + x^2 \), knowing how to square negative numbers is essential to correctly evaluating expressions and functions.

Remember, the square of a number always yields a non-negative result, which is a beneficial property when analyzing graphs and solving equations.

Evaluating Expressions

Evaluating expressions involves performing all indicated operations to simplify an expression to a value. It is a necessary step when solving mathematical problems. In our scenario, the function \( y = 1 + x^2 \) needed to be evaluated at \( x = -3 \).Start with the expression derived from substitution, \( f(-3) = 1 + (-3)^2 \). After substitution and squaring, the expression becomes \( 1 + 9 \).

• Perform Operations: Combine or simplify operations as per the order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
• Complete the Evaluation: In our example, with just addition remaining, we complete the calculation by adding the numbers: \( 1 + 9 = 10 \).
• Verification: Confirm your result by rechecking each step ensures accuracy, particularly when evaluating more complex expressions.

Evaluating expressions allows us to determine the exact value that a function or equation represents at a specific point, providing solutions to many mathematical questions.