*22/09/2021*

A. What is a one-to-one function?

A function ff is 11 -to- 11 if no two elements in the domain of ff correspond to the same element in the range of ff . In other words, each xx in the domain has exactly one image in the range. And, no yy in the range is the image of more than one xx in the domain.

If the graph of a function ff is known, it is easy to determine if the function is 11 -to- 11 . Use the Horizontal Line Test. If no horizontal line intersects the graph of the function ff in more than one point, then the function is 11 -to- 11 .

B. What is an inverse function

An inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, i.e., g(y) = x if and only if f(x) = y. The inverse function of f is also denoted as f^{-1}.

C. Mathematicians that contributed to the study of one to one function and inverse function

-LEONHARD EULER

introduced the familiar notation for the value of a function.

The functions considered in those times are called today differentiable functions

-LOBACHEVSKY and DIRICHLET

Nikolai Lobachevsky and Peter Gustav Lejeune Dirichlet are traditionally credited with independently giving the modern "formal" definition of a function as a relation in which every first element has a unique second element.

-NICOLE ORESME

Oresme invented a type of coordinate geometry before Descartes, finding the logical equivalence between tabulating values and graphing them in De configurationibus qualitatum et motuum. He proposed the use of a graph for plotting a variable magnitude whose value depends on another variable.

-Karl Weierstrass, in full Karl Theodor Wilhelm Weierstrass, (born Oct. 31, 1815, Ostenfelde, Bavaria [Germany]—died Feb. 19, 1897, Berlin), German mathematician, one of the founders of the modern theory of functions.

His domineering father sent him to the University of Bonn at age 19 to study law and finance in preparation for a position in the Prussian civil service. Weierstrass pursued four years of intensive fencing and drinking and returned home with no degree. He then entered the Academy of Münster in 1839 to prepare for a career as a secondary school teacher. At Münster he came under the influence of Cristof Gudermann, professor of mathematics, who was particularly interested in the theory of elliptic functions. Gudermann cultivated Weierstrass’s interest in the theory of functions with emphasis on the expansion of functions by power series.

D. What are the significant applications of one to one function and inverse function?

If a horizontal line intersects the graph of the function in more than one place, the functions is NOT one-to-one. HORIZONTAL LINE TEST: A function f is one-to-one and has an inverse function if and only if no horizontal line intersects the graph of f at more than one point.

E. How to you find the domain and range of inverse function?

To find the domain and range, we simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain. To calculate the range of the function, we simply express x as x=g(y) and then find the domain of g(y).

F. How do you graph inverse function?

STEP 01:Sketch both graphs on the same coordinate grid.

STEP 02:Replace f (x) with y.

STEP 03:Interchange x and y

STEP 04:solve for the new y from the equation In step 2

STEP 05.:Replace the new y with f-1 (x) if the inverse is a function.

G. One to one function examples in real life situation.

The children's have a trip to the zoo. The amount trip to the zoo is Php100. They are 25 children's in the class. How many children can go to the zoo if their budget is Php350?

Solution:

Step 1: x = number of children's

Step 2: y = total cost

Step 3: y = 100+25x

Step 4: If y = 350

Step 5: 350 = 100+25x

Step 6: To solve for x move the 100 to the other side.

Step 7: 350 -100 = 25x

Step 8: 250 = 25x

Step 9: 25 = 25

Step10: 10 = x

Step11: There are only 10 children's that can go to the zoo.