Problem 57 - Algebra A Combined Function

Question

For each function, find the indicated values. \(f(x)=-5 ;\) a. \(f(2)\) b. \(f(0)\) c. \(f(606)\)

Step-by-Step Solution

Answer

All values: -5.

Step 1: Understanding the Function

The given function is a constant function written as \(f(x) = -5\). In a constant function, the output value remains the same, no matter what the input \(x\) is.

Step 2: Calculating \(f(2)\)

For the first part, we need to find \(f(2)\). Since the function is constant, \(f(x) = -5\), the output for \(x = 2\) is \(f(2) = -5\).

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

Next, we find \(f(0)\). Like before, because \(f(x) = -5\) for any \(x\), it follows that \(f(0) = -5\).

Step 4: Calculating \(f(606)\)

Finally, we find \(f(606)\). Applying the function rule, \(f(x) = -5\), it shows that \(f(606) = -5\) as well, because the function is constant and always equals \(-5\).

Key Concepts

Evaluating Functions

Evaluating functions is a key skill in mathematics that involves finding the output of a function for a given input. This process helps us understand how a function behaves with different inputs. In the case of a constant function like \( f(x) = -5 \), evaluating is very straightforward. For every input \( x \), the function will always output the same value:

• If \( x = 2 \), then \( f(2) = -5 \).
• If \( x = 0 \), then \( f(0) = -5 \).
• If \( x = 606 \), then \( f(606) = -5 \).

This is because the function does not depend on the input value; it is constant. This makes evaluating such functions quick and simple as no matter what number you plug in for \( x \), the answer will always be \(-5\).

Function Notation

Understanding function notation is essential in grasping how functions communicate input and output values. When you see \( f(x) \), this represents "the function \( f \) of \( x \)." Think of \( x \) as a placeholder or variable where you can input any number.
The function \( f(x) = -5 \) is written in function notation, indicating that no matter what \( x \) you choose, the output will always be \(-5\). This notation helps in clearly defining what the function does without having to write out lengthy explanations each time.
When you find, say, \( f(2) \), it means you substitute \( 2 \) in place of \( x \) in the function. Although in this constant function, \( x \) does not alter the output, the notation is important for more complex functions where the output might change with different inputs.

Algebra Concepts

Algebra involves working with variables and equations to find unknown values. Understanding algebra concepts is crucial when working with functions as they often involve expressions that include variables such as \( x \).
In the problem we are discussing, the equation \( f(x) = -5 \) is an example of a constant function within algebra. This means:

• No matter what number you replace \( x \) with, the output will always remain the same because the expression does not change.
• Constant functions simplify algebraic computation since they do not require solving or simplifying equations beyond recognizing the constant output.

This also highlights the predictable nature of such functions, offering reliability in computations and allowing for straightforward problem-solving techniques in algebra.