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
NOT INVERTIBLE because I plugged in -0.5 and got a positive answer, so the derivative is then increasing and decreasing, right? The function also has two inflection points at x=0 and x=-1
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.
Let's start with b, the answer of which is correct: it is not invertible.
Domain of the function is ℝ since it is the exponential function raised to a polynomial power.
If it were invertible, as you mentioned, the derivative would not change sign.
Change in sign of the derivative implies that on each side of the extremum, the function will take on the same value within its domain, which renders the function not invertible. In other words, it does not pass the horizontal line test.
For part a.
Again, we have to first determine the domain of the function:
Since log functions cannot take on negative values, we determine that the domain of the function is limited to the range of x where the expression inside the log is non-negative, namely (-&infin,-1)∪(0,∞).
Now we have to look at the function within its domain.
We find that the function is monotonically decreasing throughout its domain, within which it satisfies the horizontal line test.
Would you therefore re-evaluate your answer?
To help make your decision, you can have some thoughts on two things:
1. look at the graph of the function:
http://img853.imageshack.us/i/1300926177.png/
2. Find its inverse:
f-1(x)=1/(10^x-1)
which is a perfectly legitimate function undefined at x=0.
To determine whether a function is invertible, you can use the definition of invertibility, which states that a function is invertible if and only if it is both one-to-one and onto.
A function is one-to-one if it passes the horizontal line test, meaning that no two points on the graph of the function have the same y-coordinate. In other words, for any two distinct x-values, the corresponding y-values must be different.
A function is onto if every y-value in the codomain (or the range of the function) has at least one corresponding x-value in the domain.
Now, let's analyze the functions you provided:
a) y = log10(1 + 1/x)
To check if this function is one-to-one, we can take its derivative. You correctly found the derivative as y' = ((1/(1 + 1/x))*ln(10)) * (-1/x^2). However, to determine if the function is one-to-one, we need to consider the sign of the derivative.
In this case, the derivative is always negative since both ln(10) and (-1/x^2) are negative. Therefore, the function is strictly decreasing, so no two distinct x-values will have the same y-value. Hence, the function y = log10(1 + 1/x) is one-to-one.
To check if the function is onto, we need to analyze the range. As x approaches positive or negative infinity, the log10(1 + 1/x) approaches 0. Therefore, the range of this function is (0, ∞). Since every positive number has a corresponding x-value (for example, log10(2) has a corresponding x-value of 1), the function is onto.
Since the function is both one-to-one and onto, it is invertible.
b) y = e^(x^2 - 5x + 6)
To check if this function is one-to-one, we can again take its derivative. You correctly found the derivative as y' = e^(x^2 - 5x + 6) * (2x - 5).
However, in this case, the derivative can be positive or negative depending on the value of x. This means that there can be two distinct x-values that correspond to the same y-value, violating the one-to-one property. Thus, the function is not one-to-one.
Since the function is not one-to-one, it cannot be invertible.
In conclusion, the function y = log10(1 + 1/x) is invertible, while the function y = e^(x^2 - 5x + 6) is not invertible.