MarkFL said:Okay, we require:
\(\displaystyle x^2-5x>0\)
Can you factor the expression on the LHS?
RidiculousName said:\(\displaystyle x^2-5x>0 \) becomes \(\displaystyle x(x-5)>0\) or \(\displaystyle x^2>5x\) depending on what I do. I'm just not sure where to take it after that.
MarkFL said:Observing that \(x^2-5x=x(x-5)\) tells us that the roots are:
\(\displaystyle x\in\{0,5\}\)
Rather than testing intervals though, let's use what we know about the parabolic graphs of quadratic functions. We see the coefficient of the squared term is positive, which means the parabola opens upwards, and so, given that it has two real roots, we should expect the expression to be positive on either side of the two roots, and negative in between. Can you proceed?
RidiculousName said:So, since the coefficient of the squared root is positive, I can tell it's \(\displaystyle (\infty,0)\cup(5,\infty)\)?
The domain of a function refers to the set of all possible input values for which the function is defined. In this case, since the natural logarithm function ln(x) is only defined for positive values of x, the domain of the given function is all real numbers greater than 0, except for the values of x that make the expression x^2−5x equal to 0. This can be written as: Domain: (0, 5) U (5, ∞).
No, the function cannot have negative input values. As mentioned earlier, the natural logarithm function is only defined for positive values of x, so any negative input values would result in an undefined output.
To find the domain of a function with a natural logarithm and an exponent, you need to consider the restrictions of both functions. In this case, the natural logarithm function has a domain of all positive real numbers, and the exponent function x^2−5x has a domain of all real numbers. Therefore, the domain of the given function is the intersection of these two domains, which results in the domain mentioned in the answer to the first question.
No, the domain of a natural logarithm function can never be negative. As mentioned earlier, the natural logarithm function is only defined for positive values of x, so any negative values would result in an undefined output.
The coefficient of x does not have any effect on the domain of a natural logarithm function. The only thing that affects the domain of a natural logarithm function is the sign of the expression inside the logarithm. As long as the expression inside the logarithm is positive, the function will be defined and its domain will remain the same.