Problem 37 - Calculus with Concepts in Calculus

Question

Determine the range of the function. $$ f(x)=-1 $$

Step-by-Step Solution

Answer

The range of the function is \(-1\).

Step 1: Identify the Function

The function given is a constant function: \( f(x) = -1 \). This means that no matter what value of \( x \) we input, the output will always be \(-1\).

Step 2: Understand the Range of a Constant Function

For a constant function, the range consists only of the constant value. Since \( f(x) = -1 \) for all \( x \), the output or range doesn't vary.

Step 3: Determine the Range

As we've established that \( f(x) \) always equals \(-1\), this means the range of the function is the set containing only this value.

Key Concepts

Constant Functions

Constant functions are some of the simplest functions you can encounter in mathematics. They don't change their output, no matter what the input is. For example, the function given in the problem, \( f(x) = -1 \), is a classic constant function. Here, the output is always \(-1\), whether \( x \) is \(-10\), \( 0 \), or \( 1000 \). The graph of a constant function is a horizontal line on the coordinate plane. This line remains steady because there's no change in the output values. Constant functions are easy to work with since they don't require you to think about different outputs—they have only one constant result. Think of constant functions like a light bulb that can only shine at one brightness level.

Identifying Functions

Identifying a function can sometimes be tricky, but with constant functions, it is very straightforward. A function in basic terms is a rule that assigns each input to exactly one output. The function \( f(x) = -1 \) clearly follows this rule.To identify functions:

• Look at the equation or rule it uses.
• Examine if each input gives you exactly one output.

For constant functions like our example, this task is quite simple. Since any input leads to the same output (in this case, \(-1\)), they're immediately recognizable. Remember, the simplicity of constant functions will help you solidify your understanding of more complex functions down the line.

Function Output

The output of a function refers to the value you get after applying the function's rule to an input. With constant functions, this concept becomes crystal clear. Take our example \( f(x) = -1 \): here, the output will always and only be \(-1\), no matter what \( x \) is.When determining the range, which is the collection of possible outputs of a function, constant functions make the process straightforward. The range for \( f(x) = -1 \) is simply \{-1\}, because that's the only value you ever get.Knowing outputs in functions help in graphing and understanding the behavior of functions. With constant functions, since the output never changes, you only have one point to mark on the output axis, no matter how many different inputs you try!