Problem 19 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(f(x)=5\) a. \(f(9)\) b. \(f(-9)\) c. \(f(0)\)

Step-by-Step Solution

Answer

The results are \(f(9)=5\), \(f(-9)=5\) and \(f(0)=5\). No matter what the x value is, the function value remains 5 since it is a constant function.

Step 1: Evaluate the function at \(x = 9\)

For the given constant function \(f(x)=5\), if \(x=9\), then \(f(9)=5\). This is because the constant function wouldn't be affected by the value of x.

Step 2: Evaluate the function at \(x = -9\)

Similarly for \(x=-9\), we will have \(f(-9)=5\). Again, the negative value of x doesn't change the output because it's a constant function.

Step 3: Evaluate the function at \(x = 0\)

Finally for \(x=0\), the function will still evaluate to 5, hence \(f(0)=5\).

Key Concepts

Constant Functions

When we talk about constant functions, we refer to a type of function in algebra where the output value does not depend on the input value. In mathematical terms, a constant function can be written as f(x) = c, where c is a fixed number, and no matter what value of x (the input) you put into the function, the output will always be c.

Think of a constant function as a flat line on a graph. It doesn't matter how far along the x-axis (horizontal direction) you go, the line never goes up or down because the y-value (vertical direction) remains the same. This property of constant functions makes them predictably simple, which is helpful when starting to learn about more complex types of functions in algebra.

Function Evaluation

The process of determining the output of a function for a particular input is known as function evaluation. In essence, when you 'plug in' a value into the function in place of the variable, the function provides the corresponding output according to its rule or equation. In our simple exercise above with f(x) = 5, evaluating the function for any input, be it 9, -9, or 0, yielded the same result: 5.

This concept is foundational in algebra as it reinforces the idea of a function being a rule that assigns exactly one output to each input. It is important to follow the function's rule precisely and be aware that even though constant functions always yield the same output, other types of functions will produce different outputs depending on the input.

Introductory Algebra

In the world of mathematics, introductory algebra is the starting point for understanding how to work with variables and functions. It introduces concepts such as solving equations, function evaluation, and graphing linear relationships. One of the key lessons in introductory algebra is recognizing different types of functions and how they behave. Constant functions are an excellent starting place because of their simplicity.

Handling these initial concepts with ease paves the way for more advanced topics like quadratic functions, exponentials, and beyond. Remember, algebra is like building with blocks. You start with the simple ones like constant functions and then stack more complex ideas on top as your understanding and skills grow.