Find values of inverse functions from tables

Learning Objectives Define the relationship between a function and its inverse by swapping input and output values. Evaluate an inverse function, f⁻¹(y), for a specific value using a table of values for f(x). Solve composite functions involving inverses, such as f(g⁻¹(x)), using data from two separate tables. Determine if a function is invertible by analyzing its table of values for one-to-one correspondence. Apply the formula for the derivative of an inverse function to find (f⁻¹)'(a) using a table that includes values for f(x) and f'(x). Interpret the meaning of an inverse function's value within a real-world context presented in a table. How does a temperature conversion app instantly switch from Celsius to Fahrenheit? 🌡️ It uses the same principle as inver...

TermDefinitionExample FunctionA mathematical rule that assigns exactly one output value for each unique input value.If f(x) = x², the input x=3 has exactly one output, f(3)=9. Inverse Function (f⁻¹)A function that reverses the action of another function. If a function f takes an input 'a' to an output 'b', its inverse f⁻¹ takes 'b' back to 'a'.If f(x) = 2x, then f(3) = 6. The inverse function is f⁻¹(x) = x/2, and f⁻¹(6) = 3. One-to-One FunctionA function where every output value is associated with exactly one input value. A function must be one-to-one to have an inverse.f(x) = x³ is one-to-one. However, f(x) = x² is not, because the output 4 corresponds to two inputs, x=2 and x=-2. DomainThe complete set of possible input values (typically x-values)...

The Fundamental Inverse Relationship If f(a) = b, then f⁻¹(b) = a This is the core principle. To find the value of an inverse function for a given input, you treat that input as an *output* of the original function and find the corresponding *input*. Composition of Inverses f(f⁻¹(x)) = x and f⁻¹(f(x)) = x When a function and its inverse are composed, they cancel each other out, returning the original input value, provided the value is in the appropriate domain. Derivative of an Inverse Function (f⁻¹)'(a) = \frac{1}{f'(f⁻¹(a))} This crucial calculus formula allows you to find the derivative of an inverse function at a point 'a' by using the derivative of the original function, f', evaluated at the point f⁻¹(a).