Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function. For a function f(x), its inverse f^-1(x) satisfies the equation f(f^-1(x)) = x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y. Understanding how to derive and interpret inverse functions is crucial for solving problems involving them.
Recommended video:
Graphing Logarithmic Functions
Exponential Functions
Exponential functions are of the form f(x) = a * b^x, where a is a constant, b is the base, and x is the exponent. In this case, f(x) = e^x + 10 is an exponential function shifted vertically by 10 units. The properties of exponential functions, such as their growth behavior and asymptotic nature, are essential for determining their inverses and understanding their domains and ranges.
Recommended video:
Domain and Range
The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For the function f(x) = e^x + 10, the domain is all real numbers, as e^x is defined for every x. The range, however, starts from 10 (the minimum value) and extends to infinity, reflecting the vertical shift of the exponential function.
Recommended video:
Domain & Range of Transformed Functions