Problem 7 - Applied Calculus

Question

For the functions, find \(f(5)\). $$ f(x)=2 x+3 $$

Step-by-Step Solution

Answer

\(f(5) = 13\)

Step 1: Substitute the value of x with 5

To find \(f(5)\), we need to replace the variable \(x\) with 5 in the function \(f(x) = 2x + 3\). This means we will calculate \(f(5) = 2(5) + 3\).

Step 2: Perform the multiplication

Now, compute the multiplication: \(2 \times 5 = 10\). The equation now becomes \(f(5) = 10 + 3\).

Step 3: Add the numbers to obtain the final result

Finally, add the remaining values: \(10 + 3 = 13\). Therefore, \(f(5) = 13\).

Key Concepts

Substitute Variable

Function evaluation often involves substituting a specific value into the variable of the function. It means that wherever you see the variable in the equation, you'll replace it with a given number. In this exercise, we have the function \( f(x) = 2x + 3 \). To find \( f(5) \), we substitute the variable \( x \) with \( 5 \). This substitution transforms our function into \( 2(5) + 3 \).

• Identify the variable in the function equation (in this case, \( x \)).
• Replace the variable with the desired numerical value (here, \( 5 \)).

This substitution process allows us to evaluate the function at any given point. It’s a crucial step that bridges the variable expression into a numerical one.

Linear Function

A linear function is a mathematical expression where the highest power of the variable is one. These functions create straight lines when graphed. The function provided in this exercise, \( f(x) = 2x + 3 \), is a typical example of a linear function.

• The coefficient of \( x \), which is \( 2 \), dictates the slope or steepness of the line.
• The constant term, \( 3 \), represents the y-intercept where the line crosses the y-axis.

Understanding linear functions helps in predicting values and trends, as they display a constant rate of change. In algebra, it’s essential to recognize linear functions, as they form the basis for more complex equations and models.

Algebraic Computation

Algebraic computation involves performing arithmetic operations like multiplication, addition, and subtraction to solve expressions. Once we substitute the variable in a function, the next step is calculation. This involves arithmetic processes that simplify the expression to a single value.In our function, after substituting \( 5 \) for \( x \), we perform:

• Multiplication: \( 2 \times 5 = 10 \).
• Addition: \( 10 + 3 = 13 \).

Each arithmetic step sequentially simplifies the expression, leading to the final result. This systematic method of computation is fundamental in algebra, making sure each action follows the rules to achieve the correct outcome. Mastering these operations is key to tackling more challenging algebraic problems.