Problem 20 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(f(x)=7\) a. \(f(10)\) b. \(f(-10) \quad\) c. \(f(0)\)

Step-by-Step Solution

Answer

The value of the function \(f(x)\) for all given values is \(7\). Thus, \(f(10) = f(-10) = f(0) = 7\).

Step 1: Evaluate \(f(10)\)

Substitute \(x = 10\) into the function \(f(x) = 7\). As this is a constant function, regardless of what value \(x\) takes, the outcome will still be 7. So, \(f(10) = 7\).

Step 2: Evaluate \(f(-10)\)

Substitute \(x = -10\) into the function \(f(x) = 7\). Here again, regardless of the value of \(x\), the outcome remains the same. Therefore, \(f(-10) = 7\).

Step 3: Evaluate \(f(0)\)

Substitute \(x = 0\) into the function \(f(x) = 7\). As the function is a constant one, the value remains at 7, even for \(x = 0\). Hence, \(f(0) = 7\).

Key Concepts

Function Evaluation

When we talk about evaluating functions, we mean finding the value of a function for a particular input. In simple terms, it's like asking: "What do I get if I put this into the function?"
In the context of constant functions like \(f(x) = 7\), the process is very straightforward.
For any input value that you choose, the output remains the same because the function always equals 7. This makes constant functions incredibly predictable and easy to evaluate.

• If your input is 10, then \(f(10) = 7\).
• If your input is -10, then \(f(-10) = 7\).
• If your input is 0, then \(f(0) = 7\).

No matter the input, the constant function "sticks" to its value.

Algebraic Functions

Algebraic functions encompass a wide variety of functions you might encounter in math. At their core, they are built from basic operations like addition, subtraction, multiplication, and division, along with powers and roots.
Constant functions, like \(f(x) = 7\), fall under the umbrella of algebraic functions. However, they are on the simpler end of the spectrum because they do not involve any variable terms altering the output.
An algebraic function may look something like \(g(x) = 3x + 2\), where the output depends on \(x\). However, in a constant function, like our \(f(x) = 7\), there's no such dependency, making them unique among algebraic functions. This uniqueness simplifies tasks like evaluation and substitution, as shown in our example.

Substitution in Functions

Substitution is a method used to evaluate a function by replacing its variable with a given value or number. This technique is integral to solving most mathematical problems involving functions.
In constant functions like \(f(x) = 7\), substitution is incredibly simple. Whatever number you substitute for \(x\), the function remains the same because it doesn't change with the input value.
Below's the consistent outcome no matter the substitution:

• Substitute 10 for \(x\) and get \(f(10) = 7\).
• Substitute -10 and get \(f(-10) = 7\).
• Substitute 0 for \(x\) and get \(f(0) = 7\).

This consistency demonstrates the often overlooked simplicity and beauty of constant functions. No replacement adjusts the result, showing that substitution in this scenario has no effect on the outcome.