Use the Desmos Graphing Calculator tool for this question. Note that whenever you are asked to produce a graph using Desmos, your screenshot must include the spot where you input your equations and points. https://www.desmos.com/calculator PART A. [2 MARKS]. State the domain and range of
using interval notation. PART B. [2 MARKS]. Determine the inverse function,
. PART C. [3 MARKS]. What is the domain and range of the inverse and how do we know it immediately? PART D. [2 MARKS]. Use function composition to prove the functions are inverses. PART E. [2 MARKS]. Using Desmos, graph
and its inverse together on their appropriate domains. Include the line of symmetry (as a dotted line) that verifies graphs of inverse functions. [Hint: See Example 1.30 in Section 1.4].
The Correct Answer and Explanation is:
To solve this problem, I can guide you through each part and provide the necessary steps. Here’s the breakdown for each part:
PART A: State the domain and range of the function using interval notation.
To provide the domain and range, I need to know the specific function you’re working with (which isn’t mentioned). Assuming it’s a simple function like a linear, quadratic, or rational function, I can guide you based on that. Could you provide the function for part A?
PART B: Determine the inverse function.
The inverse of a function can be found by switching the dependent and independent variables and solving for the new dependent variable. This often involves solving for yyy in terms of xxx. Again, I need the function to assist with this step.
PART C: Domain and Range of the Inverse.
For a function and its inverse, the domain of the function becomes the range of the inverse, and the range of the function becomes the domain of the inverse. So, once the function’s domain and range are identified, we can directly determine those for the inverse by switching them.
PART D: Use Function Composition to Prove the Functions are Inverses.
To verify that two functions are inverses, you need to perform function composition and show that:
- f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x
- f−1(f(x))=xf^{-1}(f(x)) = xf−1(f(x))=x
This will ensure that applying one function to the other results in the identity function.
PART E: Graph the Function and Its Inverse Using Desmos.
In Desmos, you would graph the function and its inverse on their respective domains. You can also plot the line of symmetry (usually y=xy = xy=x) to verify that inverse functions are symmetric about this line.
Once I have the function, I can walk you through specific steps for each part, including using Desmos to graph it. Would you like to provide the function you’re working with?
