Problem 59 - A Graphical Approach to College Algebra

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=5, x=9$$

Step-by-Step Solution

Answer

The value of \( f(9) \) is 5.

Step 1: Understanding the Function

We are given a function defined as \( f(x) = 5 \). This means for any input \( x \), the output is always 5, regardless of the value of \( x \).

Step 2: Finding the Function Value at the Given x

Since \( f(x) = 5 \) for any \( x \), we simply need to substitute \( x = 9 \) into this function. However, since \( f(x) \) is a constant function, \( f(9) = 5 \) without needing any substitution.

Step 3: Final Conclusion

The value of the function \( f(x) \) at \( x = 9 \) is \( 5 \). This confirms that the function maintains a constant value for any input.

Key Concepts

Function Evaluation

When we talk about function evaluation, we talk about finding the output of a function given an input. It means answering the question, "What do I get when I plug this specific number into my function?" In our case, the function given is a constant function, represented as \( f(x) = 5 \). Here, no matter what number is input, the output will always be 5.

Evaluating a function normally involves substituting the given value of \( x \) into the function and calculating the result. But with a constant function, you can skip this step because it does not depend on \( x \) at all. It's like filling out a form where one of the boxes always has the same answer: "5," regardless of what questions you are asked.

Function Notation

Function notation is a way of writing a function in a neat and clear format. It's a shorthand used by mathematicians to describe a function completely—both what you input, and what you output. Typically, function notation involves writing \( f(x) \). Here, \( f \) represents the function itself, "\( f \)" could stand for function, formula or any other letter, but \( x \) always represents the input.

In our exercise, the function is denoted as \( f(x) = 5 \). This tells us two things:

• The function's name is \( f \); and
• No matter what you input, the output is always 5.

This example makes it clear that varying the input (or \( x \)) does not change the output, which is constantly set at 5. Moreover, using function notation helps in communication about functions—when you see \( f(x) \), you know immediately that it is a function with \( x \) as its variable, making it more straightforward.

Algebraic Functions

Algebraic functions encompass a broad range of functions that include mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. These functions can be as simple as the one in our exercise to more complex expressions. A constant function, like \( f(x) = 5 \), is the simplest form of algebraic function.

Why is this an algebraic function? Because even though it only outputs a single constant value for each input, it is represented algebraically. Constant functions are important building blocks in mathematics; they are linear and can be presented graphically as a horizontal line on a Cartesian plane. When a graph is drawn, every point on the line represents the function's value of 5, regardless of the \( x \)-value.

• Constant functions have a slope of 0, indicating no change as \( x \) changes.
• They serve as references for understanding variability in other types of functions.

Recognizing a constant function is easy. Just look for equations where \( x \) does not influence the outcome.