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.