The document discusses one-to-one functions and inverse functions in mathematics, explaining their definitions, graphing techniques, and significance in real-world applications. It also references the historical contributions of various mathematicians to the concept of functions and provides guidelines for finding the domain and range of functions. Additional examples illustrate the use of these concepts, including a case study on horse weight growth and temperature conversion.

1. PERFORMANCE
TASK
IN
MATH

2. ONE-TO-ONE FUNCTION
• A function for which every element of the range of the
function corresponds to exactly one element of the
domain. One-to-one is often written 1-1.
• NOTE: y = f(x) is a function if it passes the vertical line
test and the horizontal line test.

3. GRAPHING ONE-TO-ONE FUNCTION
• Construct a table of values. Simply choose a number for x
• Solve for the corresponding value of y.
• Plot the points on the graph.
• Connect the points.

4. INVERSE FUNCTION
• The function obtained by switching the x and y variables in a
function.The inverse of function f is written f^(-1)
• NOTE: The new relation obtained by reversing the x and y
values of a function is not necessarily a function itself. The new
relation is only a function if the original function is one-to-one
function.

5. Finding the inverse
•Change f (x) to y
•Interchange x and y
•Solve y In terms of x
•Change y to f^(-1)

6. Graphing inverse function
• Get first the inverse of the given function.
• Construct a table of values. Simply choose a number for x.
• Solve the corresponding value of y.
• Plot the points on the graph.
• Connect the points.

7. Significant Application of One-to-One
Function and Inverse Function
•GPS or Global Positioning System
•Concert Sears andTickets
•Make-up of DNA molecules

8. •Temperature conversion
•Elements in the PeriodicTable
•Dewey decimals system
•Birthdays

9.  Both are significant because they both help interpret
various phenomena around us. These types of functions
make sense when we grew up and know the true value of
money. They are significant because it helps us to understand
the meaning of what life is. It is said that these functions
reciprocates one another and that no matter how try to
change it both are still related, like the measurements we see
in skyscrapers and various infrastructure.

10.  It also allows us to understand the weather and that in
every part of the globe there is a specific type of weather or
climate being experienced. These functions are significant for
they help us navigate our own future and what lies before us it
the result of what we do today.

11. Mathematicians
• Gottfried Leibniz – introduced the term “function” to describe a quantity
related to a curve.
• Johann Bernoulli – started calling expressions made of a single variable
“functions” to 1698 he agreed with Leibniz that any quantity formed may be
called a function of x.
• Leonhard Euler – said that the function of a variable quantity is an analytic
expression compose In anyway whatsoever of the variable quantity and
numbers or constant quantities.
• Jean – Baptiste Joseph Fourier – claimed that an arbitrary function could
be represented by a Fourier series. He said that functions were neither
continuous nor not defined by an analytic expression.

12. • Augustin – Louis Cauchy – he said that a function is being defined by
an analytic expression or by an equation or a system of equation. He
differs from his predecessors.
• Nikolai Lobachevsky – said that the general concept of a function
requires that a function of x be defined as number given for each x and
varying gradually with x.
• Johann Peter Gustav Lejeunne Dirichlet – said that y corresponds
to each x, and more over in such a way that when x ranges
continuously over the interval a to b then y is called a continuous
function of x for this interval.
• Godfrey Harold “G.H.” Hardy – defined a function as a relation
between two variables x and y such that “to some values of x at any
rate correspond values of y”

13. Finding the Domain
Solve the equation for y in terms of x
• If y is polynomial, the domain is the set of real numbers.
• If y is a rational expression and contains an expression d(s) in the
denominator, the domain is the set of real numbers except those values of x
that make the denominator equal to zero.
• If y contains a radical expression, where n is an event natural numbers,
the domain is a set of real numbers, except those values of x that made r(x)
less than zero.

14. Finding the Range
Solve the equation for x in terms of y
• If x is polynomial, the domain is the set of real numbers.
• If x is a rational expression and contains an expression d(s) in the
denominator, the domain is the set of real numbers except those values of
x that make the denominator equal to zero.
• If x contains a radical expression, where n is an event natural numbers,
the domain is a set of real numbers, except those values of x that made r(y)
less than zero.

15. PROBLEMS

16. S.T.A.R. Laboratories conducted a research to see how fast
horses can grow from birth to its first 5 months .The team
uses the equation, f(x) = 15x + 10kg. How much weight can
the horses have every month?
Solution*
Let x be represented as months:

17. • (1) = 15(1) + 10
= 15 + 10
= 25
• (3) = 15(3) + 10
= 45 + 10
= 55• (2) = 15(2) + 10
= 30 + 10
= 40
• f(x) = 15(x) + 10

18. • (4) = 15(4) + 10
= 60 + 10
= 70
• (5) = 15(5) + 10
= 75 + 10
= 85
The horses would weigh, 25Kg, 40Kg, 55Kg, 70Kg, and
85Kg In span of 5 months.

19. Stephen missed the mid-term exam because of a flu.The
normal body of human is 37° C. However, Stephen recorded 3
different readings in a span of 3 days. On the first day, his
temperature was 37.9 °C, It increased to 38.6 °C by the second
day and it reached 39.2 °C on the third day.What would be
Stephen’s temperature readings if it were in Fahrenheit Scale?

20. oFunction Rule:
°F = 1.8(°C) + 32
• F(37.9°C) = 1.8(37.9°C) + 32
= 68.22 + 32
= 100.22 °F
• F(38.6°C) = 1.8(38.6°C) + 32
= 69.48 + 32
= 101.48 °F

21. • F(39.2°C) = 1.8(39.2°C) + 32
= 70.56 + 32
= 102.56 °F
Stephen’s temperature readings in Fahrenheit are 100.22°F,
101. 48°F , 102.56°F.

22. . End

23. GROUP 4
(11 – Kindness)
• Sombilon
• Olden
• Odulio
• Dawal
• Sohal