What is the domain and range of the following function?
f(x)= - sqrt(-lnx+2)
For the domain, I got 0<x<or equal to e^2
Is this right?
And can someone help with range?
correct
And what did you get for your range?
I'm not sure what to do for the range?
Can you help please?
is it just YER?
When you take the derivative and set it equal to zero, there is no solution.
So the function has no local max/mins
So consider the endpoints of your domain
let x --- 0 from the right
lnx will be hugely negativae, so -lnx will be hugely positive , adding 2 to it won't matter much.
so √big will be big
and -√big will be -big, so the graph approaches negative infinity along the y-axis.
other endpoint, x = e^2
y = -√(-2+2) = -√0 = 0
range: -negative infinity to 0
To find the domain of the given function, we need to consider two things: the square root function and the natural logarithm function.
For the square root function, the argument (what's inside the square root) cannot be negative. Therefore, -ln(x) + 2 ≥ 0.
To solve this inequality, we isolate the natural logarithm term:
-ln(x) ≥ -2
Next, we multiply both sides by -1, but we need to flip the inequality sign since we are multiplying by a negative number:
ln(x) ≤ 2
To remove the natural logarithm, we can exponentiate both sides with base e (Euler's number):
e^(ln(x)) ≤ e^2
Simplifying, we get:
x ≤ e^2
So, the domain of the function f(x) is 0 < x ≤ e^2.
Now let's determine the range of the function. Since the function involves a square root and a logarithm, we need to think about the restrictions on their respective domains.
The square root function always outputs a non-negative value, so the range of the square root function is y ≥ 0.
For the natural logarithm function, its domain is positive numbers only. In this case, we have -ln(x) + 2, and since the natural logarithm term appears inside the square root, we need to ensure that it is positive.
To find the range, we consider the limiting behavior of the square root and logarithm functions. As x approaches 0, the value of -ln(x) + 2 becomes infinitely large. Thus, the range of the square root function is (0, ∞).
Combining the range of the square root function (y ≥ 0) and the range of the logarithm function (0, ∞), we can conclude that the range of the given function f(x) is [0, ∞).