Final answer:
The function f(x) = 6/(x-7) fails to exist at the point x = 7, as it would result in division by zero which is undefined in mathematics.
Step-by-step explanation:
The function fails to exist when the denominator of the function becomes zero, because division by zero is undefined.
To find the values for which the function fails to exist, we need to find the values of x that make the denominator equal to zero:
x - 7 = 0
x = 7
Therefore, the function fails to exist at x = 7, since division by zero is not defined.
The student is asking about the existence of a function f(x) = 6/(x-7). This type of function is known as a rational function, which can have points where it does not exist. For rational functions, non-existence typically occurs where there is a division by zero. In the case of f(x), the function fails to exist at x = 7 because that would result in a denominator of zero, which is undefined in mathematics. Apart from x = 7, the function f(x) exists for all other real values of x.