state whether the funcftion F:R to Rdefined by f(x)=3-4x is a bijective function or not . justify your answer

To determine if the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 3 - 4x \) is bijective, we need to check if the function is both **injective** (one-to-one) and **surjective** (onto).

### 1. Checking Injectivity (One-to-One):
A function is injective if different inputs map to different outputs, i.e., if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

**Proof of Injectivity:**
Suppose \( f(x_1) = f(x_2) \).

\[
3 - 4x_1 = 3 - 4x_2
\]

Subtract 3 from both sides:

\[
-4x_1 = -4x_2
\]

Divide by \(-4\):

\[
x_1 = x_2
\]

Since \( x_1 = x_2 \) whenever \( f(x_1) = f(x_2) \), the function is injective.

### 2. Checking Surjectivity (Onto):
A function is surjective if every element in the codomain (all of \(\mathbb{R}\)) is the image of at least one element from the domain.

**Proof of Surjectivity:**
Let \( y \in \mathbb{R} \) be any element in the codomain. We need to find an \( x \in \mathbb{R} \) such that \( f(x) = y \).

Starting with:

\[
y = 3 - 4x
\]

Solve for \( x \):

\[
4x = 3 - y
\]

\[
x = \frac{3 - y}{4}
\]

Since \( x = \frac{3 - y}{4} \) is a real number for any real \( y \), every \( y \) has a corresponding \( x \). Thus, the function is surjective.

### Conclusion:
Since the function \( f(x) = 3 - 4x \) is both injective and surjective, it is **bijective**.

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Would you like more details or have any questions? Here are some related questions to consider:

1. How can you determine if a function is injective or surjective using a graph?
2. What changes if the function is defined as \( f(x) = 3 - 4x^2 \)?
3. How do bijective functions relate to inverse functions?
4. Can a function be injective but not surjective?
5. What does it mean graphically for a function to be bijective?

**Tip:** To prove bijectivity, always separately check injectivity and surjectivity.

State whether the function F: R to R defined by f(x) = 3 - 4x is a bijective function or not. Justify your answer.

Determine if the function f(x) = 3 - 4x is bijective by checking its injectivity and surjectivity. Learn the definitions and proofs of injective and surjective functions with step-by-step explanations, suitable for students in Grades 11-12.

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