Problem 45 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$f(-3 k)$$

Step-by-Step Solution

Answer

The simplified result of \(f(-3k)\) is \(15k + 2\).

Step 1: Identify the given function and the value to substitute

We are given the function \(f(x) = -5x + 2\), and we want to substitute the value \(x = -3k\) into this function.

Step 2: Substitute the value

Now substitute the value \(x = -3k\) into the function \(f(x) = -5x + 2\): $$f(-3k) = -5(-3k) + 2$$

Step 3: Simplify the expression

Simplify the arithmetic operation in the expression: $$f(-3k) = 15k + 2$$ So, the simplified result of \(f(-3k) = 15k + 2\).

Key Concepts

Function Substitution

Function substitution is an important technique in algebraic functions, making it easier to work with complex expressions by replacing a variable with a specific value or expression. Imagine function substitution as a simple plug-and-play method where you replace every instance of a variable in a given function with the value or expression you have. In our scenario, we have the function \(f(x) = -5x + 2\). What we do is take a specific expression, which is \(-3k\) in this case, and substitute it wherever we see \(x\) in the function. This changes the function to \(f(-3k) = -5(-3k) + 2\). Here are the basic steps for function substitution:

• Identify the function you are working with.
• Choose the value or expression to substitute.
• Replace every instance of the original variable with this new value or expression.

Don't forget that substitution is often the first step before simplifying the function.

Simplifying Expressions

Once the substitution is done, simplifying expressions comes into play. Simplification involves reducing an expression to its simplest form for easier understanding and to reveal its underlying pattern or behaviour. In our function, after substituting \(-3k\) into \(f(x)\), we have \(f(-3k) = -5(-3k) + 2\). The objective here is to simplify this expression to make it clean and concise.For our example:

• Start by handling the multiplication within the expression. Multiply \(-5\) by \(-3k\), which results in \(15k\). Remember, two negatives make a positive when multiplied.
• Add any remaining constants such as \(+2\) in our case to the product. Hence, the expression becomes \(15k + 2\).

Utilizing these steps can make solving algebraic expressions efficient and effective.

Linear Functions

Linear functions are a fundamental concept in algebra, represented by functions where the variable's highest power is one. They have the general form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. In our function \(f(x) = -5x + 2\), the function is linear because the exponent on \(x\) is one.Key characteristics of linear functions include:

• The graph of a linear function is a straight line.
• The slope \(m\) determines the steepness of the line — positive slopes rise, while negative slopes fall.
• The \(y\)-intercept \(b\) is the point where the line crosses the \(y\)-axis.

Understanding these properties helps in grasping how changes in the values of \(m\) and \(b\) would affect the graph and the behavior of the function. In practical problems, recognizing and using the properties of linear functions simplifies the process of solving for variables.