91Ó°ÊÓ

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 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 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.