Problem 23 - Algebra Form and Function

Question

For what values of \(k\) does the equation \(-1=k\) have a solution? What does your answer tell you about the range of the function \(f(x)=-1 ?\)

Step-by-Step Solution

Answer

Answer: The equation -1=k has a solution only for a single value of k, which is -1.

Step 1: Solve for k

To solve the equation \(-1=k\), we don't need to do any manipulations, because the equation is already in the form of "equals k". Thus, we can see that \(k=-1\).

Step 2: Interpret the solution in terms of function range

Since we determined that \(k=-1\), this tells us the equation \(-1=k\) has a solution only for a single value of \(k\), which is -1. This implies that the range of the function \(f(x)=-1\) is a single value, {-1}. The function \(f(x)=-1\) is a constant function, always outputting the value -1, so it has a limited range.

Key Concepts

Constant Function

A constant function is a type of function in mathematics where the output or result is the same, no matter what the input value is. In function notation, this would look like \(f(x) = c\), where \(c\) is a constant number.
In the context of the given problem, our function is \(f(x) = -1\), meaning that whatever value of \(x\) you choose, the output will always be \(-1\).

• There is no variability in the output.
• The graph of a constant function is always a horizontal line, located at the constant value on the vertical axis.
• Because the output does not change, the range of a constant function is just that single constant.

Understanding constant functions helps in identifying that no matter what input is given, the outcome remains unchanged and fixed.

Solving Equations

Solving equations involves finding the value or values that make an equation true. In the context of the constant function equation presented, \(-1 = k\), the solving process is straightforward.
Here's how you look at it:

• Check if both sides of the equation present equal values.
• The right side of the equation, \(-1\) is already explicitly known, so the solution is directly \(-1\).
• No further modifications or calculations are required.

In more complex equations, several steps like isolating variables, combining like terms, or inverse operations might be needed, but this simple constant function requires only identification.
This exercise illustrates that for a constant function such as \(f(x) = -1\), solving for \(k\) will always be finding the constant itself.

Function Notation

Function notation is a way to express a function using symbols, typically showing how one quantity depends on another. It is often written in the form \(f(x)\). Here, \(f\) represents the function, while \(x\) stands for the input variable.
With function notation,

• The expression \(f(x) = -1\) clearly indicates that the function produces an output of \(-1\) for any given input \(x\).
• This notation does not only provide clarity but also concisely conveys the relationship between variables.
• Function notation is particularly helpful when dealing with complex functions, allowing mathematicians to describe functions without ambiguity.

Understanding function notation is crucial as it helps decode and work with mathematical expressions comprehensively, effectively analyzing both simple and complex relationships between variables.