Understanding the Range of Square Root Functions in Mathematics

What best describes the range of the function f(x) = √(ax + b)?

• A. All real numbers
• B. 0 to infinity
• C. Any real number below 0
• D. Negative values only

Answer: (B)

The function f(x) = √(ax + b) describes a square root function, which inherently has specific properties related to its output (the range). The square root function is defined only for non-negative values of its input, meaning that for the function to yield real numbers, the expression ax + b must be greater than or equal to zero. If we consider the nature of the square root, we see that it produces outputs that start at zero (when ax + b = 0) and extend upward towards positive infinity as the value of ax + b increases. Hence, the lowest value that f(x) can output is 0, and there is no upper limit to how large the output can be. This makes the range of the function specifically the interval from 0 to positive infinity. Therefore, the best description of the range of the function f(x) = √(ax + b) is indeed from 0 to infinity, reflecting the fact that the square root function can never yield negative outputs or values smaller than zero.

When tackling the function f(x) = √(ax + b), the question of its range often comes into play. So, what’s the range of this square root function? Well, it boils down to one simple fact: the outputs of this function range from 0 to infinity. Let’s break this down a bit further.

First off, square root functions have some unique properties. Unlike other functions, a square root can only produce non-negative outputs. It's like trying to find your way through a dense forest—you can only step on solid ground, no matter how tangled the branches are. You can’t step into the negatives! So, for our function to make sense as a real number, the expression ax + b must be greater than or equal to zero.

Imagine we’re analyzing the equation more closely: if ax + b equals zero, that's the moment f(x) equals zero. That's the starting point—the minimum value of our function. From there, as ax + b increases, f(x) climbs higher and higher without limits—the output can keep going up into the positive infinity. Easy peasy, right?

To make it crystal clear, let’s look at the options presented.

• A. All real numbers: Nope, because we can’t have negative outputs.

• B. 0 to infinity: Bingo! This is the correct answer.

• C. Any real number below 0: Not applicable since square roots don't dive into negatives.

• D. Negative values only: Sorry, that’s a dead end.

In essence, the best description of the range of the function f(x) = √(ax + b) is indeed from 0 to infinity. This reflects the nature of the square root function itself—it’s designed that way! As you gear up for the Ohio Assessments for Educators (OAE) Mathematics, having a firm grasp on functions like these can make all the difference.

Keep in mind the connection between input and output—it's fundamental to understanding any function you encounter in math. Just like the stars can only shine bright in a clear night sky, square roots can only present solutions within their specified range. This blend of logic and creativity in math is what makes it such an engaging subject to explore!

So whether you're studying late into the night or practicing equations on the weekend, remember this main point: for f(x) = √(ax + b), we can say with confidence that its range stretches from 0 to infinity.