علاقات والدوال العكسية

Relations and Inverse Functions

• Inverse functions are functions where the input and output values are swapped.
• A relation is a set of ordered pairs, inverse relations swap the input and output values of the ordered pairs.
• A function is a special type of relation where each input value has only one output value.
• A function is invertible (has an inverse function) if and only if it is a one-to-one function. This means each output corresponds to only one input.
• To find the inverse of a function, swap the x and y variables in the function's equation and then solve for y.
• The inverse of a function can be denoted by f⁻¹(x).
• The graph of an inverse function is a reflection of the graph of the original function across the line y = x.
• Composing a function with its inverse results in the identity function.
• If two functions are inverses of each other, the composition of the functions will yield the identity function.
• The inverse of a function can be visually identified by the horizontal line test. If a horizontal line intersects the graph of a function in more than one point, then the function is not invertible. If only one intersection, then the function is invertible.
• Identifying inverse functions can be achieved through composition.

Examples of Inverse Functions

• Example 1: If f(x) = 2x - 5, f⁻¹(x) = (x + 5)/2
• Example 2: If f(x) = x² + 1, then f⁻¹(x) = ±√(x - 1) (in this case, the inverse is not a function, as it doesn't pass the horizontal line test).

Determining Inverse Functions

• Check for Invertibility: A function f(x) has an inverse function f⁻¹(x) only if each output corresponds to a unique input. Verify this with the horizontal line test.
• Finding the Inverse Function: Swap x and y values, and solve the resulting equation for the new 'y'.

Composition of Functions

• Composition of functions is where the output of one function is used as the input of another function
• For the Inverse function (f°f⁻¹)(x)=x

Further Information

• The concept of a relation and its inverse, and the associated concept of a function and its inverse.
• The identification of inverse functions.
• Determining if a function has an inverse through the horizontal line test.
• Finding the equation of the inverse function.
• Composition of functions, which is important when determining if a function and its inverse are related.