Explain how the graph of f(x) = ln x can be used to obtain the graph of g(x) = e^(x-2).
I assume the exercise is based on the fact that you know what the graph of f(x) = lnx looks like
sketch fx) = ln x , remember x > 0
the inverse of f(x) = lnx is
h(x) = e^x
We can sketch the inverse of a function by simply reflecting it in the line y = x
suppose we have y = ln(x-2), it would simply be
f(x) translated to the right 2 units.
sketch that.
now reflect y = ln(x-2) in the line y=x and you will have g(x) = e^(x-2)
To obtain the graph of g(x) = e^(x-2) from the graph of f(x) = ln x, follow these steps:
Step 1: Start with the graph of f(x) = ln x. The graph of f(x) is a curve that passes through the point (1, 0) and has a vertical asymptote at x = 0. It increases as x increases and approaches positive infinity.
Step 2: Apply the transformation (x-2) to f(x). This shift moves the graph horizontally to the right by 2 units. So, every point on the graph is shifted 2 units to the right. For example, the point (1, 0) on the graph of f(x) will now be at (3, 0) on the graph of g(x).
Step 3: Apply the exponential function e^x to the graph obtained in step 2. This exponential function raises e (approximately 2.718) to the power of each corresponding x-value from step 2. So, if a point (x, y) was obtained in step 2, it will now be (x, e^y) on the graph of g(x).
Step 4: Apply the vertical shift of 2 units upward to the graph of g(x) obtained in step 3. This shift moves the entire graph 2 units upward. So, every y-value on the graph is increased by 2 units. For example, if a point on the graph of g(x) has a y-coordinate of 1, it will now have a y-coordinate of 3.
By following these steps, the graph of f(x) = ln x can be transformed to obtain the graph of g(x) = e^(x-2).
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 understand the properties of the natural logarithm function (ln) and the exponential function (e^x).
The natural logarithm function (ln):
The graph of f(x) = ln x is a smooth curve that passes through the point (1, 0) and increases infinitely as x approaches positive infinity. It is important to note that the natural logarithm function is only defined for x > 0.
The exponential function (e^x):
The graph of g(x) = e^x is an upward-sloping curve that passes through the point (0, 1) and increases rapidly as x increases. The base of this exponential function is the mathematical constant e ≈ 2.71828.
Now, let's explore how we can use the graph of f(x) = ln x to obtain the graph of g(x) = e^(x-2):
1. Shift the graph horizontally:
The graph of g(x) = e^(x-2) can be obtained by shifting the graph of f(x) = ln x two units to the right.
To shift the graph, we adjust the argument of the function. Instead of ln x, we use ln (x-2). This transformation shifts the graph horizontally two units to the right.
2. Reverse the effect of the horizontal shift:
To reverse the effect of the horizontal shift, we use the inverse function of ln, which is the exponential function base e.
For g(x) = e^(x-2), we rewrite it as g(x) = e^((x-2) ln e) = (e^ln e)^(x-2) = e^(x-2).
This transformation eliminates the horizontal shift, giving us the graph of g(x) = e^(x-2).
To summarize:
The graph of g(x) = e^(x-2) can be obtained from the graph of f(x) = ln x by shifting the graph of ln x two units to the right and then applying the inverse function of ln, which is e^x.
Keep in mind that these transformations maintain the overall shape of the graph while shifting and adjusting it horizontally, resulting in the desired graph.