Question

version 1 1. using the verbal description below, write the original function and its inverse. let n = starting number (input) and r = resulting number (output) starting with a number, subtract 1 from that number take the square root of that quantity add 3 r = ____________ 2 (original) n = __________ 2 (inverse) 2. graph the inverse of the function that is graphed below. (original function) (inverse) determine the following for the original function and inverse that are graphed above. domain: __________ 1 domain: __________ 1 range: __________ 1 range: ____________ 1 is the inverse relation also a function? yes or no (circle one) 1 justify your answer. 1

Explanation:

Step1: Write the original function

We start with \(n\), subtract 1 then take the square - root and add 3. So the original function \(r = \sqrt{n - 1}+3\), where \(n\geq1\) (because the expression under the square - root must be non - negative).

Step2: Find the inverse function

First, we start with \(r=\sqrt{n - 1}+3\).
Subtract 3 from both sides: \(r - 3=\sqrt{n - 1}\).
Square both sides: \((r - 3)^2=n - 1\).
Then \(n=(r - 3)^2+1\), and the domain of the original function's range (\(r\geq3\)) becomes the domain of the inverse function.

Step3: For graphing the inverse

The graph of the inverse of a function \(y = f(x)\) is obtained by reflecting the graph of \(y = f(x)\) about the line \(y = x\). To find key points on the original graph and then swap their \(x\) and \(y\) coordinates for the inverse graph.

Step4: Determine domain and range

For the original function \(r=\sqrt{n - 1}+3\), the domain is \(n\geq1\) and the range is \(r\geq3\). For the inverse function \(n=(r - 3)^2+1\), the domain is \(r\geq3\) and the range is \(n\geq1\).

Step5: Check if the inverse is a function

Since the original function passes the horizontal line test (any horizontal line intersects the graph of the original function at most once), its inverse passes the vertical line test. So the inverse relation is a function.

Answer:

1. \(r=\sqrt{n - 1}+3\) (original), \(n=(r - 3)^2+1\) (inverse)
2. (Graph is obtained by reflecting the original graph about \(y = x\))

• Original function: Domain: \(x\geq1\), Range: \(y\geq3\)
• Inverse function: Domain: \(x\geq3\), Range: \(y\geq1\)
• Is the inverse relation also a function? YES. Justification: The original function passes the horizontal line test, so its inverse passes the vertical line test.