Category: Mathematics

One of the most famous formulas in mathematics is the Pythagorean Theorem. It is based on a right triangle and states the relationship among the lengths of the sides as a2+ b2= c2, where a and b refer to the legs of a right triangle and c refers to the hypotenuse. It has immeasurable uses in engineering, architecture, science, geometry, trigonometry, algebra, and in everyday applications. For your first post, search online for an article or video that describes how the Pythagorean Theorem can be used in the real world. Provide a one paragraph summary of the article or video in your own words. Be sure you cite the article and provide the link.

Classify the following functions as one of the types of functions that we have discussed.
Quadratic Function
Find an equation of the quadratic below:
Find a formula for a cubic function if (−2) = 𝑓(1) = 𝑓(3) = 0 and 𝑓(2) = 8
f(x) = -x^3 + 6x^2 – 13x + 8
Ex 1) Let f(x) = 2x – 3. Find
a) f(–5)
b) f(2)a) f(–5) = 2(-5) – 3
= -10 – 3
= -13
b) f(2) = 2(2) – 3
= 4 – 3
= 1
Ex 2) Use the vertical line test to identify graphs in which y is a function of x.
a) b) c) d)
a) Yes – The vertical line test passes.
b) No – The vertical line test fails.

c) Yes – The vertical line test passes.
d) No – The vertical line test fails.
Ex 3) Determine whether the relations are functions:
b) c)) 2𝑥 + 𝑦2 = 16 𝑦 = 𝑥2 + 1 𝑦 = 𝑐𝑜𝑠𝑥
a) This is a function because it produces a unique output for each input. The domain is all real numbers, and the range is all real numbers greater than or equal to 4.
b) This is a function because it produces a unique output for each input. The domain is all real numbers, and the range is all real numbers greater than or equal to 1.
c) This is a function because it produces a unique output for each input. The domain is all real numbers, and the range is all real numbers greater than or equal to -1.
Ex 4) Let and𝑓 ( ) = 𝑥+1 𝑥−2 𝑔 𝑥( ) = 𝑥 + 2
Find the domain of each function.
f(x) = (x+1)/(x-2)
Domain of f(x) = {x | x ≠ 2}
g(x) = x + 2
Domain of g(x) = All real numbers
Ex 5) Let f(x) = 2x – 3. State the domain of the function, and find:

a) f(–5)
b) f(2)
c) f(a +1)
d) f(x + h)
Domain: x ∈ R
a) f(–5) = 2(-5) – 3 = -13
b) f(2) = 2(2) – 3 = 1
c) f(a + 1) = 2(a + 1) – 3 = 2a + 1 – 3 = 2a – 2
d) f(x + h) = 2(x + h) – 3 = 2x + 2h – 3
Ex 6) Simplify the difference quotient for 𝑓 𝑥+ℎ( )−𝑓(𝑥)
ℎ 𝑓 ( ) = 3 − 𝑥2
Piecewise Functions:
𝑓 𝑥+ℎ( )−𝑓(𝑥) = (3 − (𝑥+ℎ)2) − (3 − 𝑥2)
= (3 − 𝑥2 − 2𝑥ℎ − ℎ2) − (3 − 𝑥2)
= −2𝑥ℎ − ℎ2
Simplified Difference Quotient:
𝑓 𝑥+ℎ( )−𝑓(𝑥) = −2𝑥ℎ − ℎ2

Ex 7) Graph the piecewise function 𝑓 ( ){𝑥2 − 2𝑥 + 1 𝑓𝑜𝑟 𝑥 < 2 𝑥 − 1| | 𝑓𝑜𝑟 𝑥 ≥2

For x < 2
f(x) = x2 – 2x + 1
f(0) = 0 – 0 + 1 = 1
f(1) = 1 – 2 + 1 = 0
f(2) = 4 – 4 + 1 = 1

For x ≥ 2
f(x) = x – 1
f(2) = 2 – 1 = 1
f(3) = 3 – 1 = 2
f(4) = 4 – 1 = 3