Functions

In Exercises 1–6, ...

Question

Functions

In Exercises 1–6, find the domain and range of each function.

F(x) = √(5x + 10)

Accepted Answer

Welcome back, everyone. Find the domain and range of the following function F of X is equal to square root of 2X minus 8. Let's begin with the domain of the function, and we have to recall that the domain is an interval of x values for which our function is defined. Our function is F of X equals square root of 2X minus 8. And we have to recall that in real plane we cannot take square root of a negative number, meaning we want to set a restriction such that 2 x minus 8 is greater than or equal to 0. Now let's solve this inequality. First of all, we can divide both sides by 2, which gives us X minus 4 is greater than or equals to 0, and then we can add 4 to both sides, right? This gives us X is greater than or equal to 4, and this is our domain. We have shown that to make sure that we're not seeking square root of a negative number, X must be greater than or equal to 4. So for the domain, X belongs to the interval. From 4 up to infinity, this is what X is greater than or equal to 4 tells us. So let's label our first answer for the domain. And now for the range, those are the Y values that our function can acquire from minimum to maximum. What we want to simply recall is that if we plot square root of X, So let's show that we're plotting square root of X. It is basically a function that keeps increasing with an increase in x values, right? So it starts from X equals 0 and Y equals 0, and then it increases indefinitely. In this case, we simply have a transformation. We can see that 2 X minus 8 is increasing, right? Because it is simply a line with a positive slope. So essentially we can plot that line or basically have a rough idea of what that line looks like because it has two Xs with a slope of 2, which is positive. Our line. I Increasing in its y value with an increase in X. So, basically what we have shown is that for a linear function to X minus 8. It keeps increasing infinitely. So basically we can sell that the range of our function will have a minimum value at the endpoint of the interval, where basically the lower value, which is 4, right? We can state that we're evaluating F at. 4, which is square root of 2 multiplied by 4 minus 8, which is 0, so the minimum value that we can get for our F of X is 0, and the maximum value is going to be infinity, right, because we're going to take square root of an infinitely large number at the upper bound of the domain, right? And therefore we can say that the range is y belongs to the interval from zero inclusive. That's the minimum value that we can get, and the maximum value is going to be infinity, right, with an increase in x square root of X increases as well, and those will be our answers to this problem. Thank you for watching.

Functions

In Exercises 1–6, find the domain and range of each function.

F(x) = √(5x + 10)

Key Concepts

Domain of a Function

Range of a Function

Square Root Function

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