Video Transcript
Find the domain of the function 𝑓 of 𝑥 equals the square root of 𝑥 plus three plus the cube root of 𝑥 minus seven.
Remember, the domain of a function is the set of possible inputs of the function that will ensure all outputs are real. When finding domains of functions, there is two things that we need to be really aware of. Firstly, if we’re dealing with a rational function, that’s a function that’s written as the quotient of two functions, we need to ensure that the denominator of our function is not equal to zero. Similarly, if we’re evaluating the square root of a function, we need to ensure that that function is not a negative number. It can be greater than or equal to zero.
So, in our case, we need to ensure the function inside our square root is greater than or equal to zero. That is, 𝑥 plus three is greater than or equal to zero. We’ll solve this inequality by subtracting three from both sides. And we find 𝑥 is greater than or equal to negative three.
Are there any other restrictions on our domain? Well, no. We know that we can evaluate the cube root of both positive and negative numbers. So, we need to make no restrictions on our function 𝑥 minus seven. And so, we use the following interval notation to demonstrate the domain of our function. We know 𝑥 must be greater than or equal to negative three and less than ∞. We say less than rather than less than or equal to ∞ and use this curly bracket because we can’t actually evaluate 𝑥 plus three, for example, when 𝑥 is ∞. And so, the domain of our function 𝑓 of 𝑥 is values of 𝑥 greater than or equal to negative three and less than ∞.
