Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Suppose \( y = 3t + 5 \) and \( x = 21t^2 \). Write a function \( f \) that determines \( y \) in terms of \( x \). (Assume \( t \geq 0 \).)
**Solution:**
To express \( y \) as a function of \( x \), follow these steps:
1. **Solve for \( t \)** using the equation \( x = 21t^2 \):
\[
t^2 = \frac{x}{21}
\]
2. **Express \( t \)** in terms of \( x \) by taking the square root, considering \( t \geq 0 \):
\[
t = \sqrt{\frac{x}{21}}
\]
3. **Substitute \( t \)** in the equation \( y = 3t + 5 \) to express \( y \) in terms of \( x \):
\[
y = 3\left(\sqrt{\frac{x}{21}}\right) + 5
\]
Thus, the function \( f(x) \) that determines \( y \) in terms of \( x \) is:
\[
f(x) = 3\sqrt{\frac{x}{21}} + 5
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a7e6ad5-0eb5-476a-8034-bd3421cbec45%2Fb0af7c3c-475f-4fd9-8b0d-476386ac1afc%2Feq7ofv8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose \( y = 3t + 5 \) and \( x = 21t^2 \). Write a function \( f \) that determines \( y \) in terms of \( x \). (Assume \( t \geq 0 \).)
**Solution:**
To express \( y \) as a function of \( x \), follow these steps:
1. **Solve for \( t \)** using the equation \( x = 21t^2 \):
\[
t^2 = \frac{x}{21}
\]
2. **Express \( t \)** in terms of \( x \) by taking the square root, considering \( t \geq 0 \):
\[
t = \sqrt{\frac{x}{21}}
\]
3. **Substitute \( t \)** in the equation \( y = 3t + 5 \) to express \( y \) in terms of \( x \):
\[
y = 3\left(\sqrt{\frac{x}{21}}\right) + 5
\]
Thus, the function \( f(x) \) that determines \( y \) in terms of \( x \) is:
\[
f(x) = 3\sqrt{\frac{x}{21}} + 5
\]
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