conangrayisbae
Answered

Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

Given the functions:
[tex]\[ f(x)=|x| \][/tex]
[tex]\[ g(x)=x+1 \][/tex]

(a) [tex]\[ f \circ g \][/tex]
[tex]\[\square\][/tex]

(b) [tex]\[ g \circ f \][/tex]
[tex]\[\square\][/tex]

Find the domain of each function and each composite function. Enter your answers using interval notation.

Domain of [tex]\( f \)[/tex]:
[tex]\[\square\][/tex]

Domain of [tex]\( g \)[/tex]:
[tex]\[\square\][/tex]

Domain of [tex]\( f \circ g \)[/tex]:
[tex]\[\square\][/tex]

Domain of [tex]\( g \circ f \)[/tex]:
[tex]\[\square\][/tex]

Sagot :

GinnyAnswer
Let's start by breaking down each part of the composite functions as well as determining the domains.

### Composite Functions:

#### (a) [tex]\( f \circ g \)[/tex]
The notation [tex]\( f \circ g \)[/tex] means [tex]\( f(g(x)) \)[/tex].

Given [tex]\( g(x) = x + 1 \)[/tex], we need to evaluate [tex]\( f \)[/tex] at [tex]\( g(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x + 1) \][/tex]

Since [tex]\( f(x) = |x| \)[/tex], we substitute [tex]\( x + 1 \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(x + 1) = |x + 1| \][/tex]

Thus:
[tex]\[ f \circ g (x) = |x + 1| \][/tex]

#### (b) [tex]\( g \circ f \)[/tex]
The notation [tex]\( g \circ f \)[/tex] means [tex]\( g(f(x)) \)[/tex].

Given [tex]\( f(x) = |x| \)[/tex], we need to evaluate [tex]\( g \)[/tex] at [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(|x|) \][/tex]

Since [tex]\( g(x) = x + 1 \)[/tex], we substitute [tex]\( |x| \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(|x|) = |x| + 1 \][/tex]

Thus:
[tex]\[ g \circ f (x) = |x| + 1 \][/tex]

### Evaluating Composite Functions at a Given Point:

Evaluating these composite functions at [tex]\( x = 2 \)[/tex]:
- For [tex]\( f \circ g (2) \)[/tex]:
[tex]\[ f(g(2)) = f(2 + 1) = f(3) = |3| = 3 \][/tex]

- For [tex]\( g \circ f (2) \)[/tex]:
[tex]\[ g(f(2)) = g(|2|) = g(2) = 2 + 1 = 3 \][/tex]

So, the evaluations are:
[tex]\[ f \circ g (2) = 3 \][/tex]
[tex]\[ g \circ f (2) = 3 \][/tex]

### Domains:

The domain of a function [tex]\( f \)[/tex] is the set of all [tex]\( x \)[/tex] for which the function is defined.

- Domain of [tex]\( f(x) = |x| \)[/tex]:
- The absolute value function [tex]\( f(x) = |x| \)[/tex] is defined for all real numbers.
- Therefore, the domain of [tex]\( f \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].

- Domain of [tex]\( g(x) = x + 1 \)[/tex]:
- The linear function [tex]\( g(x) = x + 1 \)[/tex] is also defined for all real numbers.
- Therefore, the domain of [tex]\( g \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].

- Domain of [tex]\( f \circ g (x) = |x + 1| \)[/tex]:
- Since both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are defined for all real numbers, [tex]\( f(g(x)) \)[/tex] is defined for all real numbers.
- Therefore, the domain of [tex]\( f \circ g \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].

- Domain of [tex]\( g \circ f (x) = |x| + 1 \)[/tex]:
- Similarly, since both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are defined for all real numbers, [tex]\( g(f(x)) \)[/tex] is defined for all real numbers.
- Therefore, the domain of [tex]\( g \circ f \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].

### Summary:

- The evaluations of the composite functions at [tex]\( x = 2 \)[/tex] are:
[tex]\[ f \circ g (2) = 3 \][/tex]
[tex]\[ g \circ f (2) = 3 \][/tex]

- Domains:
- Domain of [tex]\( f \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Domain of [tex]\( g \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Domain of [tex]\( f \circ g \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
- Domain of [tex]\( g \circ f \)[/tex]: [tex]\( (-\infty, \infty) \)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.