Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
![The text provides a calculus task involving the function \( f(x) = \ln(x^2 + 1) \). Here is the transcription for educational purposes:
1. Let
\[
f(x) = \ln(x^2 + 1)
\]
Note:
\[
f'(x) = \frac{2x}{x^2 + 1} \quad \text{and} \quad f''(x) = \frac{2(1-x)(1+x)}{(x^2 + 1)^2}
\]
Tasks:
(a) Find domain
(b) Find all intercepts
(c) Draw number line indicating where \( f \) is \( + \) or \( - \) or 0 or dne
(d) Draw number line indicating where \( f' \) is \( + \) or \( - \) or 0 or dne
(e) Find all local extremes and where they occur
(f) Draw number line indicating where \( f'' \) is \( + \) or \( - \) or 0 or dne
(g) Find all inflection points
(h) Find all asymptotes
(i) Sketch a graph of \( f \) indicating local extremes and inflection points.
2. Sketch a graph of a function \( f \) that satisfies ALL number lines (one graph, not three separate).
There are no graphs or additional diagrams included in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61dec6a4-dd58-413c-ac38-185006e7681f%2F8c9d8b93-d919-4e82-8aea-42ed302ab009%2F5mgsq2o_processed.jpeg&w=3840&q=75)
Given,
(f) To draw the number line indicating where is +ve or -ve or 0.
(g) To find the inflection points.
(h) To find all the asymptotes.
Step by step
Solved in 4 steps with 1 images
