Function Properties: Analyzing f(x) = 6 - 5x

The question asks us to determine the nature of the function \(f(x) = 6 - 5x\), where \(f : R \to R\). Here, \(R\) represents the set of all real numbers. To classify this function, we need to examine two key properties: whether it is a one-to-one function (injective) and whether it is an onto function (surjective).

One-to-One Function (Injective) Analysis

A function \(f : A \to B\) is considered one-to-one, or injective, if every distinct element in the domain \(A\) maps to a distinct element in the codomain \(B\). In simpler terms, if \(f(x_1) = f(x_2)\), then it must imply that \(x_1 = x_2\).

Let's apply this definition to our given function \(f(x) = 6 - 5x\):

• Assume \(f(x_1) = f(x_2)\) for any \(x_1, x_2 \in R\).
• Substitute the function definition: \(6 - 5x_1 = 6 - 5x_2\).
• Subtract 6 from both sides of the equation: \(-5x_1 = -5x_2\).
• Divide both sides by -5: \(x_1 = x_2\).

Since we started with \(f(x_1) = f(x_2)\) and logically concluded that \(x_1 = x_2\), the function \(f(x) = 6 - 5x\) is indeed a one-to-one function. This means no two different input values will produce the same output value.

Onto Function (Surjective) Analysis

A function \(f : A \to B\) is considered onto, or surjective, if every element in the codomain \(B\) has at least one corresponding element in the domain \(A\). This means that for any \(y \in B\), there exists at least one \(x \in A\) such that \(f(x) = y\).

Let's check if our function \(f(x) = 6 - 5x\) is onto for the codomain \(R\):

• Let \(y\) be any arbitrary real number in the codomain \(R\).
• We need to find an \(x \in R\) (domain) such that \(f(x) = y\).
• Set \(f(x) = y\): \(6 - 5x = y\).
• Now, solve for \(x\) in terms of \(y\):
  • Subtract 6 from both sides: \(-5x = y - 6\).
  • Multiply by -1: \(5x = 6 - y\).
  • Divide by 5: \(x = \frac{6 - y}{5}\).

Since \(y\) is a real number, \(6 - y\) will also be a real number, and dividing a real number by 5 will always result in another real number. Therefore, for every real number \(y\) in the codomain, we can find a corresponding real number \(x\) in the domain such that \(f(x) = y\). This confirms that the function \(f(x) = 6 - 5x\) is an onto function.

Function Classification

Because the function \(f(x) = 6 - 5x\) is both one-to-one (injective) and onto (surjective), it is classified as a bijective function. A bijective function establishes a perfect one-to-one correspondence between the elements of its domain and codomain.

Final Answer Determination

Based on our analysis, the function \(f(x) = 6 - 5x\) is both one-to-one and onto.

Therefore, the correct classification for the function \(f(x) = 6 - 5x\) is "one to one and onto function".