Determine the domain of the function 𝑓 of 𝑥 equals the cubed root of four 𝑥 plus three.
The domain of a function is the set of all values on which the function acts. Or we can think of this as the set of all input values to the function. If no other restrictions are specified, then the domain of a function will be all real values for which the function is defined. So, we start with the entire set of real numbers, which we denote as ℝ, and then we consider any exclusions which need to be removed.
In this question, the function 𝑓 is a composite function. It is the cube root of the linear function four 𝑥 plus three. Now, the domain and, in fact, also the range of the cubed root function 𝑔 of 𝑥 equals the cube root of 𝑥 are each the entire set of real numbers. In other words, the cube root function does not impose any restriction on its domain. We can see this if we consider the graph of the cube root function. It is defined for all values on the 𝑥-axis, and its graph also extends to cover all values on the 𝑦-axis.
We see that, unlike the square root function, it is possible for the cube root function to act on negative values. This is because if we cube a negative number, for example, negative three, we get a negative answer. So, operating in reverse, we can find the cube root of any negative number, and it gives a negative result. So, the domain of the cube root function is the entire set of real numbers. But what about the function under the cube root?
Well, this is a linear function. We can think of it as ℎ of 𝑥 equals four 𝑥 plus three. And so this doesn’t have any restriction on its domain. We can calculate the value of four 𝑥 plus three for any value of 𝑥 in the real numbers. This means that there are no restrictions for the domain of the linear function and no restrictions for the domain of the cube root function. And so, overall, there are no restrictions to the possible 𝑥-values for this function. We therefore don’t need to exclude any values from the set of real numbers.
So, the domain of the function 𝑓 of 𝑥 is the complete set of real numbers, which we denote by ℝ.