The function 5x^2+2x-1's domain is all real numbers, according to the answer key. However, I don't know how to factor this to prove it. Could you show me?
The function the sq. root of y-10 has a domain of all numbers so that y is greater than or equal to zero. But I don't see how this is true. 1-10, for example would be -9, and this would be imaginary. Negative numbers would result in an imaginary value too, but I don't see how -1-10 is any more imaginary than 1-10.
There isn't an "easy" factorisation for 5x^2+2x-1, so you can use the formula for the factors of
ax^2 + bx + b
(-b +- sqrt(b^2-4ac)) /2a
but I'm not quite sure why this proves the domain.
In the second part, I think I agree with you, but it depends where you put the brackets!
sqrt(y) - 10 has a domain y >= 0
sqrt(y-10) has a domain y-10 >= 0
Might that be the difficulty here?
To determine the domain of a function, we need to analyze any limitations or restrictions on the variables involved.
Let's first consider the function 5x^2 + 2x - 1. This is a quadratic function in the form of ax^2 + bx + c. To determine the domain, we need to check if there are any restrictions on the value of x.
In this case, there are no square roots or fractions involved, which are common sources of restrictions. Since there are no limitations on x, we can conclude that the domain is all real numbers.
Now let's move on to the function √(y - 10). This is a square root function in the form of √(expression). To determine the domain, we need to identify any restrictions on the expression inside the square root.
In this case, the expression y - 10 is under the square root. To ensure the square root is defined, the expression inside it must be greater than or equal to zero. Thus, we have the inequality y - 10 ≥ 0.
To solve this inequality:
1. Add 10 to both sides of the inequality: y - 10 + 10 ≥ 0 + 10, which simplifies to y ≥ 10.
2. This means that for the square root to be defined, the value of y must be greater than or equal to 10.
So, the domain of the function √(y - 10) is all real numbers where y is greater than or equal to 10.
Regarding your specific confusion with the example of 1 - 10 and -1 - 10, let's evaluate both expressions:
1 - 10 = -9, which is a negative number.
-1 - 10 = -11, which is also a negative number.
Both of these results are indeed negative numbers. However, when substituting these values into the function √(y - 10), we need to remember that the square root of a negative number is not defined within the real number system.
So, although both -9 and -11 are negative, they are different from zero and, therefore, do not satisfy the condition of y being greater than or equal to 10. Hence, these values are not part of the domain of the function.