Problem 26 - Algebra 2

Question

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

Step-by-Step Solution

Answer

f(-5)=21, f(-3)=13, f(1/2)=-1, f(4)=-15

Step 1: Evaluate f(-5)

Substitute the variable d with -5 in the function f(d)=1-4d and simplify.

Step 2: Evaluate f(-3)

Substitute the variable d with -3 in the function f(d)=1-4d and simplify.

Step 3: Evaluate f(1/2)

Substitute the variable d with 1/2 in the function f(d)=1-4d and simplify.

Step 4: Evaluate f(4)

Substitute the variable d with 4 in the function f(d)=1-4d and simplify.

Key Concepts

Function Evaluation

Evaluating a function is like using a very specific recipe for baking: you have your ingredients, which are the values you plug into the function, and you get a result, which is the value the function gives back. In the world of mathematics, this process is called function evaluation. It means finding the value of a function for specific inputs.

For example, the exercise provided asks to evaluate the function f(d) = 1 - 4d for various values of d (in this case, -5, -3, 1/2, and 4). To do this, one simply replaces the variable d with each of these specified inputs and performs the arithmetic operation given in the function's definition.

Here, function evaluation will show us how the output of a function changes as its input changes, which is a foundational concept for understanding how functions behave.

Substitution Method

In the heart of evaluating functions lies a straightforward technique known as the substitution method. This method involves taking the value you want to evaluate and plugging it directly into the function in place of the variable.

Using our exercise as an example, if we’re looking to find f(-5), we follow these steps: identify what needs to be replaced (in this case, the variable d), and then swap it out with the value we’re evaluating (-5). So, f(d) becomes f(-5) = 1 - 4(-5).

Then, it's just a matter of simplifying the equation by carrying out the arithmetic to get your answer. This method functions as a key tool for understanding and working with all types of functions, providing a clear and intuitive path from question to answer.

Linear Functions

Our function in the exercise, f(d) = 1 - 4d, is a perfect example of what mathematicians call a linear function. Linear functions are ones where there’s just a straight-line relationship between the input and the output.

In general form, a linear function can be written as f(x) = mx + b, where m is the slope of the line and b is the y-intercept, or where the line crosses the y-axis. In our exercise, the slope is -4 and the y-intercept is 1. These functions are graphically represented by straight lines and are one of the most basic and essential concepts in algebra.

They model relationships with a constant rate of change and appear in various real-world situations, like calculating cost based on quantity or determining the total distance traveled over time at a constant speed. Understanding linear functions equips students to make sense of numerous everyday phenomena.