1. 1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Composite and Inverse Functions Translation, combination, composite Inverse, vertical/horizontal line test

2. For a positive real number c, vertical shifts of y = f(x) are: 1. Vertical shift c units upward: h(x) = y + c = f(x) + c 2. Vertical shift c units downward: h(x) = y  c = f(x)  c Vertical Shifts (rigid transformation)

3. For a positive real number c, horizontal shifts of y = f(x) are: 1. Horizontal shift c units to right: h(x) = f(x  c) ; x  c = 0, x = c 2. Vertical shift c units to left: h(x) = f(x  c) ; x + c = 0, x = -c Horizontal Shifts (rigid transformation)

4. Reflections in the coordinate axes of the graph of y = f(x) are represented as follows. 1. Reflection in the x-axis: h(x) =  f(x) (symmetric to x-axis) 2. Reflection in the y-axis: h(x) = f(  x) (symmetric to y-axis) Reflections in the Axes

5. Let x be in the common domain of f and g. 1. Sum:(f + g)(x) = f(x) + g(x) 2. Difference:(f  g)(x) = f(x)  g(x)  Product:(f  g) = f(x)  g(x) 4. Quotient: Arithmetic Combinations

6. The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function f with the function g is defined by (f u g)(x) = f(g(x)). Two step process to find y = f(g(x)): 1. Find h = g(x). 2. Find y = f(h) = f(g(x)) Composite Functions

7. One-to-One Function For y = f(x) to be a 1-1 function, each x corresponds to exactly one y, and each y corresponds to exactly one x. A 1-1 function f passes both the vertical and horizontal line tests.

8. VERTICAL LINE TEST for a Function A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

9. HORIZONTAL LINE TEST for a 1-1 Function The function y = f(x) is a one-to-one (1-1) function if no horizontal line intersects the graph of f at more than one point.

10. A function, f, has an inverse function, g, if and only if (iff) the function f is a one-to-one (1-1) function. Existence of an Inverse Function

11. A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f. Definition of an Inverse Function

12. If the function f has an inverse function g, then domainrange fxygxyfxygxy Relationship between Domains and Ranges of f and g

13. 1. Given the function y = f(x). 2. Interchange x and y. 3. Solve the result of Step 2 for y = g(x). 4. If y = g(x) is a function, then g(x) = f -1 (x). Finding the Inverse of a Function