Answered

Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

2003/8 Exercise 6.6

Which of the following functions is/are one-to-one?

[tex]\[ f: x \rightarrow x^2 \][/tex]
[tex]\[ g: x \rightarrow 2x + 1 \][/tex]
[tex]\[ h: x \rightarrow \sqrt{x} \][/tex]

A. \( f \) only
B. \( g \) only
C. \( h \) only
D. [tex]\( f \)[/tex] and [tex]\( h \)[/tex] only

Sagot :

GinnyAnswer
To determine which of the functions \( f(x) = x^2 \), \( g(x) = 2x + 1 \), and \( h(x) = \sqrt{x} \) are one-to-one, we need to analyze each function and determine if each input \( x \) maps to a unique output \( y \).

### Function \( f(x) = x^2 \)

1. Recall that a function is one-to-one if different inputs map to different outputs.
2. Let’s assume \( f(x_1) = f(x_2) \):
[tex]\[ x_1^2 = x_2^2 \][/tex]
3. This implies:
[tex]\[ x_1 = x_2 \quad \text{or} \quad x_1 = -x_2 \][/tex]
This shows that different inputs (e.g., \( x_1 = 2 \) and \( x_2 = -2 \)) can map to the same output (e.g., \( 4 \)).
4. Therefore, \( f(x) = x^2 \) is not one-to-one.

### Function \( g(x) = 2x + 1 \)

1. Assume \( g(x_1) = g(x_2) \):
[tex]\[ 2x_1 + 1 = 2x_2 + 1 \][/tex]
2. Subtracting 1 from both sides:
[tex]\[ 2x_1 = 2x_2 \][/tex]
3. Dividing both sides by 2:
[tex]\[ x_1 = x_2 \][/tex]
This shows that different inputs \( x_1 \) and \( x_2 \) must be the same if the outputs are the same.
4. Therefore, \( g(x) = 2x + 1 \) is one-to-one.

### Function \( h(x) = \sqrt{x} \)

1. Assume \( h(x_1) = h(x_2) \):
[tex]\[ \sqrt{x_1} = \sqrt{x_2} \][/tex]
2. Squaring both sides:
[tex]\[ x_1 = x_2 \][/tex]
This shows that different inputs \( x_1 \) and \( x_2 \) must be the same if the outputs are the same.
3. Therefore, \( h(x) = \sqrt{x} \) is one-to-one on its domain (non-negative real numbers).

Based on this analysis:
- \( f(x) = x^2 \) is not one-to-one (because it maps both \( x \) and \( -x \) to the same value).
- \( g(x) = 2x + 1 \) is one-to-one.
- \( h(x) = \sqrt{x} \) is one-to-one.

Therefore, the correct answer is B: [tex]\( g \)[/tex] only.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. 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.