Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Let's analyze each pair of functions step by step to determine the compositions [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex], and then check whether [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses.
## Pair (a)
Given functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
### Step 1: Calculate [tex]\( f(g(x)) \)[/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x - 3) = (x - 3) + 3 = x \][/tex]
### Step 2: Calculate [tex]\( g(f(x)) \)[/tex]
[tex]\[ f(x) = x + 3 \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 3) = (x + 3) - 3 = x \][/tex]
Since both compositions [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
## Pair (b)
Given functions:
[tex]\[ f(x) = -\frac{1}{4x}, x \neq 0 \][/tex]
[tex]\[ g(x) = \frac{1}{4x}, x \neq 0 \][/tex]
### Step 1: Calculate [tex]\( f(g(x)) \)[/tex]
[tex]\[ g(x) = \frac{1}{4x} \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{4x}\right) = -\frac{1}{4 \left(\frac{1}{4x}\right)} = -\frac{1}{\frac{1}{x}} = -x \][/tex]
### Step 2: Calculate [tex]\( g(f(x)) \)[/tex]
[tex]\[ f(x) = -\frac{1}{4x} \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g\left(-\frac{1}{4x}\right) = \frac{1}{4 \left( -\frac{1}{4x} \right)} = \frac{1}{-\frac{1}{x}} = -x \][/tex]
Since neither composition [tex]\( f(g(x)) = -x \)[/tex] nor [tex]\( g(f(x)) = -x \)[/tex] simplifies to [tex]\( x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverses of each other.
## Summary:
(a) For [tex]\( f(x) = x + 3 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex]:
[tex]\[ f(g(x)) = x \][/tex]
[tex]\[ g(f(x)) = x \][/tex]
[tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
(b) For [tex]\( f(x) = -\frac{1}{4x}, x \neq 0 \)[/tex] and [tex]\( g(x) = \frac{1}{4x}, x \neq 0 \)[/tex]:
[tex]\[ f(g(x)) = -x \][/tex]
[tex]\[ g(f(x)) = -x \][/tex]
[tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverses of each other.
## Pair (a)
Given functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
### Step 1: Calculate [tex]\( f(g(x)) \)[/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x - 3) = (x - 3) + 3 = x \][/tex]
### Step 2: Calculate [tex]\( g(f(x)) \)[/tex]
[tex]\[ f(x) = x + 3 \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 3) = (x + 3) - 3 = x \][/tex]
Since both compositions [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
## Pair (b)
Given functions:
[tex]\[ f(x) = -\frac{1}{4x}, x \neq 0 \][/tex]
[tex]\[ g(x) = \frac{1}{4x}, x \neq 0 \][/tex]
### Step 1: Calculate [tex]\( f(g(x)) \)[/tex]
[tex]\[ g(x) = \frac{1}{4x} \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{4x}\right) = -\frac{1}{4 \left(\frac{1}{4x}\right)} = -\frac{1}{\frac{1}{x}} = -x \][/tex]
### Step 2: Calculate [tex]\( g(f(x)) \)[/tex]
[tex]\[ f(x) = -\frac{1}{4x} \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g\left(-\frac{1}{4x}\right) = \frac{1}{4 \left( -\frac{1}{4x} \right)} = \frac{1}{-\frac{1}{x}} = -x \][/tex]
Since neither composition [tex]\( f(g(x)) = -x \)[/tex] nor [tex]\( g(f(x)) = -x \)[/tex] simplifies to [tex]\( x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverses of each other.
## Summary:
(a) For [tex]\( f(x) = x + 3 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex]:
[tex]\[ f(g(x)) = x \][/tex]
[tex]\[ g(f(x)) = x \][/tex]
[tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
(b) For [tex]\( f(x) = -\frac{1}{4x}, x \neq 0 \)[/tex] and [tex]\( g(x) = \frac{1}{4x}, x \neq 0 \)[/tex]:
[tex]\[ f(g(x)) = -x \][/tex]
[tex]\[ g(f(x)) = -x \][/tex]
[tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverses of each other.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
