Sure, I can certainly explain that to you.
Firstly, let's understand what domains and ranges are. The domain of a function includes all possible input values (x-values) which will yield real numbers as outputs, and the range represents all possible output values (y-values) that can be validly derived from the function.
Let's start with the function f(x) = 5x + 17.
Here, you can input any real number as x, and 5x + 17 will always output a real number. That means the domain is all real numbers which we typically represent as (-∞, ∞). This range of (-∞, ∞) means it extends from negative infinity to positive infinity, covering every possible real number.
Now, let's move to the range of f(x) = 5x + 17. No matter what real number you give as x, the function will always yield a real number because if you multiply any real number by 5 and add 17, you will always get a real number. Hence, the range is all real numbers as well, represented as (-∞, ∞).
Next, we examine the function g(x) = 4x^2.
The domain is all real numbers (-∞, ∞) because we can square any real number, and then multiply it by 4 to get a real number output. Therefore, any real number for x is valid.
Now, let's consider the range of g(x) = 4x^2. When we square any real number (either positive or negative), the result is always a positive number or zero (if x=0). So, no matter which real number we input as x, the output of 4x^2 will always be a positive number (including zero). Therefore, the range of g(x) = 4x^2 is all positive real numbers including zero, represented as (0, ∞).
Comparing both functions, f(x) = 5x + 17 and g(x) = 4x^2, we can see that both have the same domain which is all real numbers (-∞, ∞). However, their ranges are different. The range of f(x) = 5x + 17 is all real numbers (-∞, ∞), while the range of g(x) = 4x^2 is all positive real numbers including zero (0, ∞).
I hope this clarifies the domains and ranges of these functions for you!