Chemistry

Zenah Abdulaal

Dai Thai

Chem. 400

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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) = ?(1) = ?(3) = 0 and ?(2) = 8

f(x) = -x^3 + 6x^2 – 13x + 8

Ex 1) Let f(x) = 2x – 3. Find

1. a) f(–5)
2. b) f(2)a) f(–5) = 2(-5) – 3

    = -10 – 3

    = -13

1. 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.

1. a) b) c) d)
2. a) Yes – The vertical line test passes.
3. b) No – The vertical line test fails.

 

1. c) Yes – The vertical line test passes.
2. d) No – The vertical line test fails.

Ex 3) Determine whether the relations are functions:

1. b) c)) 2? + ?2 = 16 ? = ?2 + 1 ? = ????
2. 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.
3. 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.
4. 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 and? ( ) = ?+1 ?−2 ? ?( ) = ? + 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:

 

1. a) f(–5)
2. b) f(2)
3. c) f(a +1)
4. d) f(x + h)

Domain: x ∈ R

1. a) f(–5) = 2(-5) – 3 = -13
2. b) f(2) = 2(2) – 3 = 1
3. c) f(a + 1) = 2(a + 1) – 3 = 2a + 1 – 3 = 2a – 2
4. d) f(x + h) = 2(x + h) – 3 = 2x + 2h – 3

Ex 6) Simplify the difference quotient for ? ?+ℎ( )−?(?)

ℎ ? ( ) = 3 − ?2

Piecewise Functions:

? ?+ℎ( )−?(?) = (3 − (?+ℎ)2) − (3 − ?2)

= (3 − ?2 − 2?ℎ − ℎ2) − (3 − ?2)

= −2?ℎ − ℎ2

Simplified Difference Quotient:

? ?+ℎ( )−?(?) = −2?ℎ − ℎ2

 

 

 

Ex 7) Graph the piecewise function ? ( ){?2 − 2? + 1 ??? ? < 2 ? − 1| | ??? ? ≥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