Problem 13 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of the function \(f(-3)\) is 2.

Step 1: Understand the Function

The function given is an equation where \[y = f(x) = \sqrt{1-x}\] and we need to find the value of this function when \(x = -3\).

Step 2: Substitute the Value of x

Substitute \(x = -3\) into the function to find \(f(-3)\).We have \[f(-3) = \sqrt{1 - (-3)}\]

Step 3: Simplify the Expression Inside the Square Root

Calculate the expression inside the square root by simplifying \[1 - (-3) = 1 + 3 = 4\]

Step 4: Calculate the Square Root

Find the square root of the simplified expression.So, \[f(-3) = \sqrt{4} = 2\]

Key Concepts

Substitution Method

In mathematics, the substitution method is a critical technique used to evaluate functions with variable inputs. This method is especially important when working with functions where you have to find the function's value for a specific input. To begin with, you replace the variable in the function, often denoted as \( x \), with a given number. For example, in the exercise, we have \( y = f(x) = \sqrt{1-x} \) and are tasked with finding \( f(-3) \).
To accomplish this, you substitute \( x = -3 \) into the function. This replacement allows you to extract a numerical value, allowing the function to yield a specific output. The resulting expression becomes \( f(-3) = \sqrt{1 - (-3)} \). You are essentially "plugging in" the number into the function, enabling you to work towards a specific answer.

• This method can be used for different types of functions, including linear and polynomial functions, among others.
• It simplifies the problem-solving process by dealing with one equation at a time.
• Accurate substitution is crucial for arriving at the correct result, as any error in this step could lead to incorrect outcomes.

Square Root Function

A square root function is a special kind of function that involves the square root of a variable expression. When you see \( \sqrt{...} \), it indicates that you are looking for a number which, when multiplied by itself, gives the original number inside the square root.
In our exercise, the function given is \( f(x) = \sqrt{1-x} \). Understanding how the square root operation works is essential for solving and simplifying these expressions. Once the value of the expression inside the square root is determined (e.g., 1 - (-3) = 4), you then need to find the square root of that result. Here, \( \sqrt{4} \) gives us 2, because \( 2 \times 2 = 4 \).

• The function \( y = \sqrt{1-x} \) implies that only values of \( x \) making \( 1-x \geq 0 \) are possible in 'real' number solutions.
• Square root functions generally produce a curve when graphed, typically in one quadrant of the coordinate plane under normal circumstances.
• Understanding domain restrictions (values that \( x \) can take) is crucial for valid function evaluation.

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra that involves reducing expressions to their simplest form, making them easier to work with. This often involves combining like terms and performing basic arithmetic operations. When simplifying the expression inside a square root, such as \( 1 - (-3) \), straightforward arithmetic is used.
The negative sign before the \(-3\) changes the subtraction into addition, simplifying the expression to \( 1 + 3 \). This operation is crucial because it affects the value of the expression inside the square root.

• Pay attention to negatives, as incorrect handling can lead to errors in calculations.
• Simplification often involves combining terms and removing unnecessary complexity.
• The goal of simplification is to produce a cleaner, more manageable expression.

Mastering these simplification techniques ensures precision and accuracy in mathematics.