Determine whether or not each of the following functions is invertible. Give your reasons for believing the function is invertible or not.
Please check this for me. I am not sure if I am adequately explaining my answer and if my answer is right.
a) y= log10(1 + 1/x)
y'= ((1/(1 + 1/x)*ln(10)) * (-1/x^2))
I plugged 100 and -100 into the derivative and got -4.3 X 10^-5 and -4.4 X 10^-5
From this I concluded that no matter what number is plugged into the derivative, the result will be negative. Therefore, there is no maximum or minimum value. The function is only decreasing and there is only one x-value for every y-value, and is thus INVERTIBLE. The function has two inflection points at x=0 and x=-1 but it should not prevent the function from having an inverse.
b) y= e^(x^2 - 5x + 6)
y'= (e^(x^2 - 5x + 6) * (2x - 5))
I plugged 10 and -10 into the derivative and got 3.1 X 10^25 and -1.4 X 10^69
The function is increasing and decreasing, so there is a max and min value and there is more than one x-value for the y-value, thus NOT INVERTIBLE.
Correction to part a) NOT INVERTIBLE because I plugged in -0.5 and got a positive answer, so the derivative is then increasing and decreasing, right?
To determine whether a function is invertible, we need to examine its derivative and consider two main factors: monotonicity and bijectivity.
a) For the function y = log10(1 + 1/x):
First, let's find the derivative of the function:
y' = (1 / (1 + 1/x)) * (ln(10)) * (-1/x^2) = -ln(10) / (x^2 + x)
You correctly computed the derivative correctly. However, to determine invertibility, we need to analyze its properties.
Monotonicity:
By observing the derivative, we can determine the function's direction in terms of increasing or decreasing. The derivative is negative for all x-values, indicating that the function is monotonically decreasing. This means that for every y-value, there is exactly one x-value. Therefore, the function is one-to-one and satisfies the condition required for invertibility.
Inflection Points:
While you mentioned there are inflection points at x = 0 and x = -1, this doesn't affect the invertibility of the function. Inflection points only indicate when the curvature changes, but they don't prevent the function from having an inverse.
In conclusion, based on the fact that the function is monotonically decreasing and satisfies the one-to-one property, we can conclude that it is invertible.
b) For the function y = e^(x^2 - 5x + 6):
Again, let's find the derivative of the function:
y' = e^(x^2 - 5x + 6) * (2x - 5)
You computed the derivative correctly. Now let's analyze its properties.
Monotonicity:
Looking at the derivative, the sign of the derivative changes. This suggests that the function is not strictly increasing or decreasing. Consequently, some y-values may have multiple corresponding x-values. Therefore, the function fails the one-to-one property necessary for invertibility.
In conclusion, based on the fact that the function is not strictly monotonically increasing or decreasing, we can conclude that it is not invertible.
Overall, your explanations and reasoning are mostly accurate, but be sure to explicitly mention the monotonicity and bijectivity properties when determining the invertibility of a function.