Problem 23 - Algebra 2

Question

For each function, find \(f(-5), f(-3), f\left(\frac{1}{2}\right),\) and \(f(4)\) $$ f(y)=-3 y-2 $$

Step-by-Step Solution

Answer

f(-5)=13, f(-3)=7, f(1/2)=-3.5, f(4)=-14

Step 1: Evaluate f(-5)

Substitute -5 for y in the function f(y) to find f(-5). The calculation will be f(-5) = -3(-5) - 2.

Step 2: Simplify f(-5)

Multiply -3 by -5 and then subtract 2. This results in f(-5) = 15 - 2, which simplifies to f(-5) = 13.

Step 3: Evaluate f(-3)

Substitute -3 for y in the function f(y) to find f(-3). Perform the calculation f(-3) = -3(-3) - 2.

Step 4: Simplify f(-3)

Multiply -3 by -3 and then subtract 2. This results in f(-3) = 9 - 2, which simplifies to f(-3) = 7.

Step 5: Evaluate f(1/2)

Substitute 1/2 for y in the function f(y) to find f(1/2). The calculation will be f(1/2) = -3(1/2) - 2.

Step 6: Simplify f(1/2)

Multiply -3 by 1/2 and then subtract 2. This results in f(1/2) = -1.5 - 2, which simplifies to f(1/2) = -3.5.

Step 7: Evaluate f(4)

Substitute 4 for y in the function f(y) to find f(4). The calculation will be f(4) = -3(4) - 2.

Step 8: Simplify f(4)

Multiply -3 by 4 and then subtract 2. This results in f(4) = -12 - 2, which simplifies to f(4) = -14.

Key Concepts

Function Substitution

Understanding how to substitute values into a function is essential for evaluating the function at specific points. Function substitution is straightforward: you simply replace the variable in the function with the value you are interested in evaluating.

For example, if we have a function given by the equation \( f(y) = -3y - 2 \), and we want to calculate \( f(-5) \), we substitute \( y \) with \( -5 \). The substitution step looks like this: \( f(-5) = -3(-5) - 2 \).

It's like a recipe where the variable \( y \) is the ingredient, and we are just swapping it out for a different ingredient to see how it changes the outcome. This technique is a bedrock skill for algebra and makes understanding more complex topics much easier.

Simplifying Expressions

The process of simplifying expressions enables us to reduce an expression to its most basic form. In the context of function evaluation, after substituting a value into a function, we often need to perform algebraic operations to simplify the result.

Let's take the step of simplifying \( f(-5) \) after substitution: We begin with \( -3(-5) - 2 \), which simplifies to \( 15 - 2 \). This further simplifies to \( 13 \) by performing basic arithmetic operations. Here, simplification includes multiplying and then subtracting numbers.

Importance of Order

Remember to follow the order of operations (PEMDAS/BODMAS) while simplifying. This knowledge is crucial as it ensures that we simplify expressions correctly, especially when they contain multiple operations.

Linear Functions

Linear functions are some of the simplest and most studied functions in algebra. They can be recognized by their straight-line graphs and are represented by the standard form \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

The given function \( f(y) = -3y - 2 \) is a linear function wherein \( -3 \) is the slope and \( -2 \) the y-intercept. The slope indicates how steep the line is, and the intercept tells us where the line crosses the y-axis. For instance, if we evaluate this function at several points, like \( -5, -3, \frac{1}{2}, \) and \( 4 \), we can plot these points to visualize the straight line created by this function.

Understanding linear functions is essential, as they form the foundation for analyzing relationships between variables in many fields of study.