Milkesa Abrahim
19 hours ago
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ChatGPT
19 hours ago
Inverse functions are functions that "undo" the action of another function. In other words, if a function f(x) maps an input x to an output y, then the inverse function f^-1(y) maps the output y back to the original input x.
The graph of an inverse function is a reflection of the original function's graph across the line y = x. This means that the x and y coordinates of each point on the original function's graph are swapped to create the graph of the inverse function.
For example, if the original function is f(x) = x^2, then the inverse function would be f^-1(y) = √y. The graph of f(x) = x^2 is a parabola opening upwards, while the graph of f^-1(y) = √y is a curve that looks like the right half of a sideways parabola.
Inverse functions are useful in many areas of mathematics, such as solving equations, finding the roots of functions, and analyzing the behavior of functions.
The graph of an inverse function is a reflection of the original function's graph across the line y = x. This means that the x and y coordinates of each point on the original function's graph are swapped to create the graph of the inverse function.
For example, if the original function is f(x) = x^2, then the inverse function would be f^-1(y) = √y. The graph of f(x) = x^2 is a parabola opening upwards, while the graph of f^-1(y) = √y is a curve that looks like the right half of a sideways parabola.
Inverse functions are useful in many areas of mathematics, such as solving equations, finding the roots of functions, and analyzing the behavior of functions.
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Google Gemini AI
19 hours ago
Inverse Functions
An inverse function is a function that undoes the original function. In other words, if f(x) = y, then the inverse function, denoted as f^(-1)(y), will provide x for a given y.
Finding the Inverse Function
To find the inverse function of f(x), follow these steps:
1. Replace f(x) with y.
2. Interchange x and y.
3. Solve for y.
Example:
Find the inverse function of f(x) = 2x + 1.
1. f(x) = y becomes y = 2x + 1.
2. Interchange x and y: x = 2y + 1.
3. Solve for y: y = (x - 1)/2.
Therefore, the inverse function is f^(-1)(x) = (x - 1)/2.
Graphs of Inverse Functions
The graph of an inverse function is a reflection of the original function across the line y = x.
- Original function: f(x)
- Inverse function: f^(-1)(x)
- Reflection line: y = x
Properties of Inverse Function Graphs:
- The points on the graph of f(x) and f^(-1)(x) are symmetric with respect to the line y = x.
- The domain and range of f(x) become the range and domain of f^(-1)(x), respectively.
- The inverse function passes the "horizontal line test" if and only if the original function passes the "vertical line test."
Example:
The graph of the function f(x) = 2x + 1 is a line with a slope of 2 and a y-intercept of 1. Its inverse function, f^(-1)(x) = (x - 1)/2, is also a line with a slope of 1/2 but a y-intercept of -1/2. The graph of f^(-1)(x) is a reflection of the graph of f(x) across the line y = x.
An inverse function is a function that undoes the original function. In other words, if f(x) = y, then the inverse function, denoted as f^(-1)(y), will provide x for a given y.
Finding the Inverse Function
To find the inverse function of f(x), follow these steps:
1. Replace f(x) with y.
2. Interchange x and y.
3. Solve for y.
Example:
Find the inverse function of f(x) = 2x + 1.
1. f(x) = y becomes y = 2x + 1.
2. Interchange x and y: x = 2y + 1.
3. Solve for y: y = (x - 1)/2.
Therefore, the inverse function is f^(-1)(x) = (x - 1)/2.
Graphs of Inverse Functions
The graph of an inverse function is a reflection of the original function across the line y = x.
- Original function: f(x)
- Inverse function: f^(-1)(x)
- Reflection line: y = x
Properties of Inverse Function Graphs:
- The points on the graph of f(x) and f^(-1)(x) are symmetric with respect to the line y = x.
- The domain and range of f(x) become the range and domain of f^(-1)(x), respectively.
- The inverse function passes the "horizontal line test" if and only if the original function passes the "vertical line test."
Example:
The graph of the function f(x) = 2x + 1 is a line with a slope of 2 and a y-intercept of 1. Its inverse function, f^(-1)(x) = (x - 1)/2, is also a line with a slope of 1/2 but a y-intercept of -1/2. The graph of f^(-1)(x) is a reflection of the graph of f(x) across the line y = x.
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