Mathematics 1
Summary
Laura Maria Vasiliu
,Chapter 1 – Functions of One Variable
Functions of one variable
Zeros of a function
Point of intersection of two graphs
Elementary functions
Solving inequalities
Power functions
Polynomial functions
Exponential functions
Logarithmic function
A. Functions of One Variable
Definition:
A function of one variable x is a prescription y(x) which calculates a number, the function value,
for any feasible value of the variable x:
y = y(x)
Example:
Demand Function:
INPUT VARIABLE (INDEPENDENT) FUNCTION OUTPUT (DEPENDENT)
p = price of good; 0 ≤ p ≤50 d(p) = 100 – 2p d = d(10) = 80, for p=10
Domain of the Function The set of all possible function values
(If there is no explicitly given domain, then the is called the range of the function.
domain consists of all x for which the function
makes sense)
B. Zeros of a Function
Definition:
A zero of a function y(x) is a solution of the equation y(x) = 0.
2
, C. Point of Intersection of Two Graphs
Definition:
A point of intersection of the graph y(x) with the graph of another function g(x) is a point (a, b),
where: a = the solution of y(x) = g(x)
b = y(a) or b = g(a) .
○ The intersection points of two graphs are called the break-even points.
D. Elementary Functions
Function Name Function Format Zeros of the Function
Constant function y(x) = c -
b
Linear function y(x) = ax + b, a ≠ 0 x= −a
Quadratic function y(x) = ax2 + bx + c, a ≠ 0 −b ± √D
x1,x2 =
2×a
D = b2 – 4ac = The Discriminant
−b ± √D
D > 0 → Two zeros → x =
2×a
b
If: D = 0 → One zero → x= −a
D < 0 → No zeros → no solutions
E. Solving Inequalities
Definition:
In the case of an inequality f(x) ≥ g(x), there is 4 – Step Process to be followed:
Step 1: Set the inequality to 0.
Step 2: Determine the zeros of the new function.
Step 3: Make the sign chart.
Step 4: Reach the solution.
3
, Example: f(x) ≥ g(x) Step 1: h(x) = f(x) – g(x)
Step 2: h(x) = 0 → determine the zeros
h(x) a 0 -a 0 a
Step 3:
x x1 x2
sign of a sign of a
opposite sign of a
Step 4: the values where h(x) > 0 are identical to the values where f(x) ≥ g(x).
F. Power Functions
Definition:
A function of the form y(x) = xk where k = { 0, 1, 2…} → Positive Integer Power Function
degree
Using positive integer power functions, we define a Negative Integer Power Function:
1
y(x) = x-k = No zeros
xk
Properties of Power Functions:
xm × xn = xm + n (xm)n = xm × n xm × ym = (x × y)m
1 𝑚
xm x0 = 1 𝑛
𝑥 𝑛 = √𝑥 𝑚
= xm - n x-1 =
xn x
G. Polynomial Functions
Definition:
A function of the form y(x) = anxn + an – 1xn – 1 + an – 2xn – 2 + … + a1x + a0 where n is a non-
negative integer and an ≠ 0.
o “n” = the degree of the polynomial function
4
Summary
Laura Maria Vasiliu
,Chapter 1 – Functions of One Variable
Functions of one variable
Zeros of a function
Point of intersection of two graphs
Elementary functions
Solving inequalities
Power functions
Polynomial functions
Exponential functions
Logarithmic function
A. Functions of One Variable
Definition:
A function of one variable x is a prescription y(x) which calculates a number, the function value,
for any feasible value of the variable x:
y = y(x)
Example:
Demand Function:
INPUT VARIABLE (INDEPENDENT) FUNCTION OUTPUT (DEPENDENT)
p = price of good; 0 ≤ p ≤50 d(p) = 100 – 2p d = d(10) = 80, for p=10
Domain of the Function The set of all possible function values
(If there is no explicitly given domain, then the is called the range of the function.
domain consists of all x for which the function
makes sense)
B. Zeros of a Function
Definition:
A zero of a function y(x) is a solution of the equation y(x) = 0.
2
, C. Point of Intersection of Two Graphs
Definition:
A point of intersection of the graph y(x) with the graph of another function g(x) is a point (a, b),
where: a = the solution of y(x) = g(x)
b = y(a) or b = g(a) .
○ The intersection points of two graphs are called the break-even points.
D. Elementary Functions
Function Name Function Format Zeros of the Function
Constant function y(x) = c -
b
Linear function y(x) = ax + b, a ≠ 0 x= −a
Quadratic function y(x) = ax2 + bx + c, a ≠ 0 −b ± √D
x1,x2 =
2×a
D = b2 – 4ac = The Discriminant
−b ± √D
D > 0 → Two zeros → x =
2×a
b
If: D = 0 → One zero → x= −a
D < 0 → No zeros → no solutions
E. Solving Inequalities
Definition:
In the case of an inequality f(x) ≥ g(x), there is 4 – Step Process to be followed:
Step 1: Set the inequality to 0.
Step 2: Determine the zeros of the new function.
Step 3: Make the sign chart.
Step 4: Reach the solution.
3
, Example: f(x) ≥ g(x) Step 1: h(x) = f(x) – g(x)
Step 2: h(x) = 0 → determine the zeros
h(x) a 0 -a 0 a
Step 3:
x x1 x2
sign of a sign of a
opposite sign of a
Step 4: the values where h(x) > 0 are identical to the values where f(x) ≥ g(x).
F. Power Functions
Definition:
A function of the form y(x) = xk where k = { 0, 1, 2…} → Positive Integer Power Function
degree
Using positive integer power functions, we define a Negative Integer Power Function:
1
y(x) = x-k = No zeros
xk
Properties of Power Functions:
xm × xn = xm + n (xm)n = xm × n xm × ym = (x × y)m
1 𝑚
xm x0 = 1 𝑛
𝑥 𝑛 = √𝑥 𝑚
= xm - n x-1 =
xn x
G. Polynomial Functions
Definition:
A function of the form y(x) = anxn + an – 1xn – 1 + an – 2xn – 2 + … + a1x + a0 where n is a non-
negative integer and an ≠ 0.
o “n” = the degree of the polynomial function
4
