**Problem 6: Function Analysis**Let $ f(x) = x - 1 $ and $ g(x) = (x + 2)^2 $, find the following:**a) Domain of $ f(x) $:**This refers to all the possible input values (x-values) for which the function $ f(x) $ is defined.**b) Range of $ f(x) $:**This involves all the possible output values (y-values) that $ f(x) $ can produce.**c) Domain of $ g(x) $:**This refers to all the possible input values for the function $ g(x) $ that make it defined.**d) Range of $ g(x) $:**This identifies all the possible outputs that $ g(x) $ can yield.**e) $ (f-g)(x) = $**Compute the result of subtracting $ g(x) $ from $ f(x) $.**f) $ (f \cdot g)(x) = $**Calculate the product of $ f(x) $ and $ g(x) $.**g) $ (f \circ g)(x) = $**Evaluate the composition of $ f $ with $ g $, i.e., $ f(g(x)) $.**h) $ (g \circ f)(x) = $**Evaluate the composition of $ g $ with $ f $, i.e., $ g(f(x)) $.**i) $ f^{-1}(x) $ =**Determine the inverse of the function $ f(x) $.**j) Is $ g(x) $ a 1-1 function? Explain:**Explain whether $ g(x) $ is a one-to-one function, where each input leads to a unique output.**Note:** - There are no graphs or additional diagrams associated with this text. - A one-to-one function means no two different inputs produce the same output.

Answered: 6. Let f(x) = x − 1 and g(x) = (x +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem 6: Function Analysis**

Let $ f(x) = x - 1 $ and $ g(x) = (x + 2)^2 $, find the following:

**a) Domain of $ f(x) $:**

This refers to all the possible input values (x-values) for which the function $ f(x) $ is defined.

**b) Range of $ f(x) $:**

This involves all the possible output values (y-values) that $ f(x) $ can produce.

**c) Domain of $ g(x) $:**

This refers to all the possible input values for the function $ g(x) $ that make it defined.

**d) Range of $ g(x) $:**

This identifies all the possible outputs that $ g(x) $ can yield.

**e) $ (f-g)(x) = $**

Compute the result of subtracting $ g(x) $ from $ f(x) $.

**f) $ (f \cdot g)(x) = $**

Calculate the product of $ f(x) $ and $ g(x) $.

**g) $ (f \circ g)(x) = $**

Evaluate the composition of $ f $ with $ g $, i.e., $ f(g(x)) $.

**h) $ (g \circ f)(x) = $**

Evaluate the composition of $ g $ with $ f $, i.e., $ g(f(x)) $.

**i) $ f^{-1}(x) $ =**

Determine the inverse of the function $ f(x) $.

**j) Is $ g(x) $ a 1-1 function? Explain:**

Explain whether $ g(x) $ is a one-to-one function, where each input leads to a unique output.

**Note:**

- There are no graphs or additional diagrams associated with this text.
- A one-to-one function means no two different inputs produce the same output.

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The function $f (x)$ is a rule that associates or fixes a number to the particular point x. If $y = f (x),$ then the independent variable x is called the input of the function and the dependent variable y is called the output of the function.