Question

1. a student is graphing the function $f(x) = 2x + 10$. possible inputs: __________ impossible inputs: __________ what do the possible inputs have in common?

Explanation:

Step1: Analyze the function type

The function \( f(x) = 2x + 10 \) is a linear function. For linear functions of the form \( f(x)=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept), there are no restrictions on the input \( x \) from the algebraic structure of the function itself.

Step2: Determine possible inputs

Since there are no values of \( x \) that would make the function undefined (there is no division by zero, no square roots of negative numbers in the real - number system for this function), the possible inputs for \( x \) are all real numbers. In set - builder notation, we can write the possible inputs as \( \{x|x\in\mathbb{R}\} \), or we can list examples like \( - 5,0,3,100 \), etc.

Step3: Determine impossible inputs

Because the function is defined for all real numbers, there are no impossible inputs (in the context of real - number arithmetic). If we consider only real numbers, there are no values of \( x \) for which \( f(x)=2x + 10 \) is not defined.

Step4: Identify the commonality of possible inputs

The possible inputs (all real numbers) are all real numbers. They can be any number on the number line, including integers, fractions, positive numbers, negative numbers, and zero. Mathematically, they belong to the set of real numbers, and there are no restrictions based on the form of the linear function.

Answer:

Possible inputs: All real numbers (e.g., \(-2, 0, 5, \frac{1}{2}\), etc.)
Impossible inputs: None (in the real - number system)
What do the possible inputs have in common? They are all real numbers (any real number can be used as an input for the linear function \(f(x)=2x + 10\)).