Find the range of the function 𝑓 of 𝑥 is equal to the absolute value of two 𝑥 plus one.
Here, we’ve been given an absolute value function. To find the output of our function, we substitute a value of 𝑥 into the expression two 𝑥 plus one. We then make that value positive no matter what we start with. And that’s the output of our function. And there are a couple of ways we can find the range of our function. The domain is the set of possible values of the independent variable. In other words, the possible 𝑥-values that will make the function work. And the domain of an absolute value function of a polynomial is simply the set of all real numbers.
Then, we say that the range of a function is the complete set of all possible resulting values of our dependent variable after we’ve substituted the domain. In other words, the resulting 𝑦-values we get after substituting in all the possible 𝑥-values. Now, earlier, we said that when we substitute in any value of 𝑥 into the expression two 𝑥 plus one, the absolute value sign tells us to make the result positive. Well, the absolute value of two 𝑥 plus one will be zero if 𝑥 is equal to negative one-half. And for all other values of 𝑥, i.e., any value of 𝑥 which is not equal to negative one-half, we get a resulting answer that’s greater than zero. So, this means the absolute value of two 𝑥 plus one will be greater than or equal to zero for all real values of 𝑥.
Using set notation, that looks a little something like this. The range is greater than or equal to zero and less than ∞. We could have, however, considered how this might look graphically. The graph of 𝑦 equals two 𝑥 plus one is a single straight line that passes through the 𝑦-axis at one and the 𝑥-axis at negative one-half. To sketch the graph of 𝑓 of 𝑥 equals the absolute value of two 𝑥 plus one, we take the absolute value of 𝑦. So, we make all 𝑦-values positive. Essentially, the part of the line that appears below the 𝑥-axis is reflected in the 𝑥-axis, as shown. We said that the range is the resulting 𝑦-values.
We can see quite clearly from the graph that these are all values greater than or equal to zero. And so, our range is greater than or equal to zero and less than ∞.