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Magazine Chapter 2: Problem 13
Draw a machine diagram for the function. $$f(x)=\sqrt{x-1}$$
Short Answer
Expert verified
Input \( x \), subtract 1, then take the square root.
Step by step solution
01
Identify Inputs and Outputs
The function given is \( f(x) = \sqrt{x - 1} \). Identify what goes into the function (input) and what comes out (output). Here, the input is \( x \) and the output is the result of \( \sqrt{x - 1} \).
02
Verify Domain of the Function
Since the function involves a square root, the expression inside the square root \( x - 1 \) must be non-negative. Therefore, we'll need \( x - 1 \geq 0 \) which simplifies to \( x \geq 1 \). So, the domain of \( f(x) \) is all real numbers \( x \) such that \( x \geq 1 \).
03
Break Down the Function Operations
To create the machine diagram, break down the functional output process:1. Subtract 1 from the input (\( x \)).2. Take the square root of the resulting value.
04
Draw the Diagram
Now, create the diagram using boxes or arrows to show the flow:- Start with \( x \).- An arrow points to a block labeled "Subtract 1" leading to an output labeled \( x - 1 \).- Another arrow continues from \( x - 1 \) into a block labeled "Square Root".- The output of this final block is \( \sqrt{x - 1} \), which represents \( f(x) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Input-Output
When dealing with functions, it's essential to understand the idea of inputs and outputs. Think of it like a process or a "machine" where you put something in to receive something else out. Here, our function is defined as \( f(x) = \sqrt{x - 1} \). This tells us a few key things:
- The input is \( x \). This is what we put into our machine, or function.
- The output is \( \sqrt{x - 1} \). It's the final result after the function has done its work.
Domain of Square Root Function
The domain of a function is all the possible inputs that won't cause any issues within the function. For functions that involve square roots, like \( f(x) = \sqrt{x - 1} \), only non-negative numbers can be under the square root.
To find the domain:
To find the domain:
- Look at the expression under the square root \( x - 1 \).
- Ensure that \( x - 1 \geq 0 \). Solving this inequality gives \( x \geq 1 \).
Machine Diagram Steps
Creating a machine diagram is like mapping out a series of steps showing how the input transforms into an output. For our specific function \( f(x) = \sqrt{x - 1} \), imagine the following steps:
- Start with the input, \( x \).
- The first step or machine operation is to "subtract 1" from \( x \). This transforms your input to \( x - 1 \).
- The next step is to "apply the square root" function to the result. This gives us \( \sqrt{x - 1} \).
Real Numbers Domain
In mathematics, the term "real numbers" refers to a vast set of numbers that include almost any number you can think of: integers, fractions, and irrational numbers. When we define the domain of a function, like in our case \( x \geq 1 \), we specify the set of real numbers that are eligible for use as inputs.
Here鈥檚 what you need to know:
Here鈥檚 what you need to know:
- Real numbers are numbers that can be found on the number line. They don鈥檛 include imaginary numbers.
- For \( f(x) = \sqrt{x - 1} \), the domain is \( x \geq 1 \). This means real numbers such as 1, 1.5, 2, 3, and so forth can be used as inputs.
