Use the functions f(x) = x − 3 and g(x) = x³ to find the value or function. (g-¹ of ¹)(-3) (g-¹ of ¹)(-3) =
Use the functions f(x) = x − 3 and g(x) = x³ to find the value or function. (g-¹ of ¹)(-3) (g-¹ of ¹)(-3) =
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
![**Using Composite and Inverse Functions**
In this example, we will use the given functions \( f(x) = \frac{1}{8}x - 3 \) and \( g(x) = x^3 \) to find the value of \( (g^{-1} \circ f^{-1})(-3) \).
1. Find the inverse of \( f(x) \):
\[
f(x) = \frac{1}{8}x - 3
\]
Start by letting \( y = f(x) \):
\[
y = \frac{1}{8}x - 3
\]
Solve for \( x \):
\[
y + 3 = \frac{1}{8}x
\]
\[
8(y + 3) = x
\]
Therefore, the inverse function is:
\[
f^{-1}(x) = 8(x + 3)
\]
2. Apply the inverse function \( f^{-1}(x) \) to -3:
\[
f^{-1}(-3) = 8(-3 + 3)
\]
\[
f^{-1}(-3) = 8 \cdot 0 = 0
\]
3. Now find the inverse of \( g(x) \):
\[
g(x) = x^3
\]
Let \( y = g(x) \):
\[
y = x^3
\]
Solve for \( x \):
\[
x = \sqrt[3]{y}
\]
Therefore, the inverse function is:
\[
g^{-1}(x) = \sqrt[3]{x}
\]
4. Apply the inverse function \( g^{-1}(x) \) to the result of \( f^{-1}(-3) \):
\[
g^{-1}(f^{-1}(-3)) = g^{-1}(0)
\]
\[
g^{-1}(0) = \sqrt[3]{0} = 0
\]
Therefore, the value of \( (g^{-1} \circ f^{-1})(-3) \) is:
\[
(g^{-1} \circ f^{-1})(-3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2367e66-1ae4-4371-aa7b-db391deab840%2F5582f3ab-18cd-42cc-9629-216543650d39%2Fitdo2tf_processed.png&w=3840&q=75)
Transcribed Image Text:**Using Composite and Inverse Functions**
In this example, we will use the given functions \( f(x) = \frac{1}{8}x - 3 \) and \( g(x) = x^3 \) to find the value of \( (g^{-1} \circ f^{-1})(-3) \).
1. Find the inverse of \( f(x) \):
\[
f(x) = \frac{1}{8}x - 3
\]
Start by letting \( y = f(x) \):
\[
y = \frac{1}{8}x - 3
\]
Solve for \( x \):
\[
y + 3 = \frac{1}{8}x
\]
\[
8(y + 3) = x
\]
Therefore, the inverse function is:
\[
f^{-1}(x) = 8(x + 3)
\]
2. Apply the inverse function \( f^{-1}(x) \) to -3:
\[
f^{-1}(-3) = 8(-3 + 3)
\]
\[
f^{-1}(-3) = 8 \cdot 0 = 0
\]
3. Now find the inverse of \( g(x) \):
\[
g(x) = x^3
\]
Let \( y = g(x) \):
\[
y = x^3
\]
Solve for \( x \):
\[
x = \sqrt[3]{y}
\]
Therefore, the inverse function is:
\[
g^{-1}(x) = \sqrt[3]{x}
\]
4. Apply the inverse function \( g^{-1}(x) \) to the result of \( f^{-1}(-3) \):
\[
g^{-1}(f^{-1}(-3)) = g^{-1}(0)
\]
\[
g^{-1}(0) = \sqrt[3]{0} = 0
\]
Therefore, the value of \( (g^{-1} \circ f^{-1})(-3) \) is:
\[
(g^{-1} \circ f^{-1})(-3)
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