A function is one-to-one (or injective) if different inputs produce different outputs. Formally, this means that if 
f
(
a
)
=
f
(
b
)
f(a)=f(b), then it must be true that 
a
=
b
a=b.

Proof of One-to-One:

Suppose 
f
(
a
)
=
f
(
b
)
f(a)=f(b).
By definition of 
f
f, this means 
a
+
1
=
b
+
1
a+1=b+1.
Subtracting 1 from both sides gives us 
a
=
b
a=b.
Since we have shown that 
f
(
a
)
=
f
(
b
)
⇒
a
=
b
f(a)=f(b)⇒a=b, we conclude that 
f
(
n
)
=
n
+
1
f(n)=n+1 is one-to-one.

The proof you've provided is essentially showing that the function \( f(n) = n + 1 \) is one-to-one (or injective). Let's break it down clearly:

### Statement to prove:
We want to show that the function \( f(n) = n + 1 \) is one-to-one. That is, if \( f(a) = f(b) \), then we must show \( a = b \).

### Proof:

1. **Assume \( f(a) = f(b) \)**:  
   This is the starting assumption. So, we begin by assuming that the function values for two inputs, \( a \) and \( b \), are equal. That is:
   \[
   f(a) = f(b)
   \]

2. **Substitute the definition of \( f \)**:  
   The function is defined as \( f(n) = n + 1 \), so substitute this into the equation:
   \[
   a + 1 = b + 1
   \]

3. **Simplify**:  
   Now, subtract 1 from both sides of the equation:
   \[
   a = b
   \]

4. **Conclusion**:  
   Since we have shown that \( f(a) = f(b) \) leads to \( a = b \), this confirms that the function \( f(n) = n + 1 \) is indeed one-to-one (injective).

Thus, \( f(n) = n + 1 \) is a one-to-one function.

---

Would you like a further explanation or breakdown of any specific step? Let me know if you have more questions.

Here are 5 follow-up questions you might find useful:

1. What other properties must a function satisfy to be bijective?
2. How do you test whether a function is injective using its graph?
3. Can a quadratic function be one-to-one? Why or why not?
4. What’s the difference between injective and surjective functions?
5. Can a function be injective but not surjective? Can you give an example?

**Tip**: When trying to prove that a function is one-to-one, it's always useful to start with the assumption \( f(a) = f(b) \) and then manipulate the equation to show that it must lead to \( a = b \).

A function is one-to-one (or injective) if different inputs produce different outputs. Formally, this means that if f(a) = f(b), then it must be true that a = b. Proof of One-to-One: Suppose f(a) = f(b). By definition of f, this means a + 1 = b + 1. Subtracting 1 from both sides gives us a = b. Since we have shown that f(a) = f(b) ⇒ a = b, we conclude that f(n) = n + 1 is one-to-one.

Learn how to prove that the function f(n) = n + 1 is one-to-one (injective). This detailed step-by-step proof shows how different inputs lead to different outputs, confirming the injectivity of the function. Suitable for high school students in Grades 9-12 studying functions and proofs.

Math Problem Statement

A function is one-to-one (or injective) if different inputs produce different outputs. Formally, this means that if f ( a )

f ( b ) f(a)=f(b), then it must be true that a

Suppose f ( a )

f ( b ) f(a)=f(b). By definition of f f, this means a + 1

b + 1 a+1=b+1. Subtracting 1 from both sides gives us a

b a=b. Since we have shown that f ( a )

f ( b ) ⇒ a

b f(a)=f(b)⇒a=b, we conclude that f ( n )

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