the denominator cannot be zero.
So, its discriminant must be negative.
That means
1-4c < 0
f(x)=x^2+1/x^2-x+c has a domain of all real numbers? Thank you!
So, its discriminant must be negative.
That means
1-4c < 0
The denominator x^2 - x becomes zero when:
x(x - 1) = 0
This equation implies that x = 0 or x - 1 = 0.
Therefore, the function would have a zero denominator and be undefined at x = 0 and x = 1.
In order to have a domain of all real numbers, we need to make sure that these values do not cause any issues in the overall function. Since the only potential issue is division by zero, we must remove x = 0 and x = 1 from the domain of the function.
Now, let's consider the remaining parts of the function:
f(x) = x^2 + 1/(x^2 - x) + c
We want to find the smallest value of c that keeps the function defined for all real numbers except x = 0 and x = 1.
In this case, since the function is a sum of terms, the value of c does not affect the domain of the function. Thus, there is no specific smallest integer value of c that satisfies the condition of a domain of all real numbers.
Therefore, no specific integer value of c is needed, and any integer value of c would result in a function with a domain of all real numbers except x = 0 and x = 1.
First, let's look at the denominator expression, x^2 - x + c. We want to avoid any values of x that could make this expression equal to zero. To find those values, we can set the quadratic equation equal to zero and solve for x:
x^2 - x + c = 0
The discriminant of the quadratic equation is Δ = b^2 - 4ac. In this case, a = 1, b = -1, and c = c. For the quadratic expression to have real roots, the discriminant must be greater than or equal to zero, Δ ≥ 0.
(-1)^2 - 4(1)(c) ≥ 0
1 - 4c ≥ 0
-4c ≥ -1
4c ≤ 1
c ≤ 1/4
So, to avoid any values of x that would make the denominator zero, we need to select a value of c that does not exceed 1/4.
However, we also need to consider the numerator x^2 + 1. Since the square of any real number is always non-negative, x^2 + 1 will always be greater than or equal to 1. Therefore, the numerator will never cause any issues regardless of the value of c.
So the smallest integer value of c that allows the function f(x) = x^2 + 1 / (x^2 - x + c) to have a domain of all real numbers is c = 0.
Therefore, the answer is c = 0.