Explain how the graph of f(x) = ln x can be used to obtain the graph of g(x) = e^(x-2).
f^-1(x) = e^x
g(x) = f^-1(x-2)
So, reflect f(x) around the line y=x, and then shift it 2 to the right.
In the graphs at
http://www.wolframalpha.com/input/?i=plot+y%3Dln%28x%29%2C+y%3De^%28x-2%29%2C+y%3Dx%2C+y%3De^x
you can see that
f(x) is the blue curve
the green curve is f inverse
The red curve is the green curve shifted to the right
To understand how the graph of f(x) = ln x can be used to obtain the graph of g(x) = e^(x-2), we need to look at the relationship between these two functions and their properties.
1. Understanding ln(x) function:
The natural logarithm function, ln(x), is the inverse of the exponential function e^x. It gives the value of the exponent to which the base 'e' (approximately 2.71828) must be raised to obtain the number 'x'. The graph of ln(x) has a few key properties:
- Domain: The function ln(x) is only defined for x > 0 as the logarithm of a non-positive number is undefined.
- Range: The range of ln(x) is (-∞, ∞), meaning it takes all real values.
2. Understanding the relationship between ln(x) and e^x:
Since ln(x) and e^x are inverse functions, their graphs are reflections of each other across the line y = x. So, if we are given the graph of ln(x), we can obtain the graph of e^x by reflecting the graph of ln(x) across the line y = x.
However, in the given function g(x) = e^(x-2), the graph is slightly shifted horizontally by 2 units to the right. Hence, we need to adjust our approach accordingly.
3. Obtaining the graph of g(x) = e^(x-2) from the graph of f(x) = ln x:
To obtain the graph of g(x) = e^(x-2), you can follow these steps:
- Start with the graph of f(x) = ln x.
- Shift the graph 2 units to the right to account for the (x-2) term in g(x).
- Flip the graph vertically as e^x is the inverse of ln(x).
By applying these transformations, you should be able to obtain the graph of g(x) = e^(x-2). It will have a vertical shift of 2 units to the right compared to the graph of ln(x), and it will have the same general shape but mirrored across the line y = x.