Composition of Functions
f(x)=2x+1 and g(x)=x2
Find (f+g)(x)
x2+2x+1
Identify the vertex of the parabola f(x) = (x - 3)2 + 7
(3,7)
the problem is in vertex form already.
Factor 6x3 - 18x2.
(Hint: This equation is not set equal to 0)
6x2(x - 3)
What is the requirement(s) for a function to have an inverse?
it must be one-to-one
(horizontal line test)
Injective
Bijective
What is the parent function of |x| + 5?
|x|
Find (f ÷ g)(x)
where
f(x) = x2 - 4 and g(x) = x - 1
(x2 - 1) / (x - 1)
where x ≠ 1
Find the x-intercepts of f(x) = x2 - 5x + 6.
x = 2, 3
Factor or quadratic formula
Factor the 'Difference of Squares':
49x2 - 81
(7x - 9)(7x + 9)
If f(1) = 5 and f(3) = 8, find f-1(8)
3
What transformation occurs when f(x) is replaced by f(-x)?
reflection over y axis
find (g ∘ f)(x) for
f(x)=x - 5 and g(x) = x2 + 2
(g ∘ f)(x) = x2 - 10x + 27
Convert f(x) = x2 - 4x + 1 into vertex form.
f(x) = (x - 2)2 - 3
Factor: 2x2 - x = 3
(Hint: Use AC method)
(x + 1)(2x - 3)
Find the inverse of: f(x)= 3x - 7
f-1(x) = (x + 7) / 3
If f(x) = x2, write the equation for a graph shifted left 3 and down 4.
f(x) = (x + 3)2 - 4
Decompose H(x) = √(x2 + 1) into two functions
f(x) and g(x)
such that
H(x) = f(g(x)).
f(x) = √x and g(x) = x2 + 1
Find the x-intercepts of f(x) = x2 - 4x +13
x = 2 ± 3i
Need to use quadratic formula
Factor: 3x2 + 7x = 6
(3x - 2)(x + 3)
Find the inverse of f(x) = √x-2 and state the domain.
f-1(x) = x2 + 2
x ≥ 0
When you take the inverse of a function, the domain and range are swapped.
(in the original function, the range is ≥ 0 and domain is ≥ 2)
Describe all transformations in h(x) = -2√(x - 1) +3
reflected over x-axis
Vertical stretch by 2
right 1
up 3
A clothing store offers a $10 discount and a 15% off coupon that can be used together. If the store applies the discount first and then the coupon, write the composite function and find the price of a $100 jacket.
(hint: $10 discount is: d(x) = x - 10 where x is the original item price)
$10 discount = d(x) = x - 10
15% off = c(x) = 0.85x
c(d(x)) = 0.85(x-10) = $76.50
A ball is thrown upward; its height is modeled by h(t) = -16t2 + 64t + 5.
What is the maximum height the ball reaches?
69 feet at t = 2 seconds
Vertex (a is negative, so vertex is maximum)
Give the following in factored form:
f(x) = -0.01x2 + 20x - 500
(hint: there is an easy way and a hard way to do this)
(x - 25.3206)(x - 1974.68)
The easy way is to use the quadratic formula (calculator). Take the outputs and put them in factored form. The hard way is anything else.
Show if the following are inverse of each:
f(x) = 1 / (x + 1)
g(x) = (1 - x) / x
yes
you can verify by taking (g ∘ f) or (f ∘ g) to get x back.
Or
you can verify by taking the inverse of both to find they're equal to the other.
If the graph of f(x) = √x is horizontally compressed by a factor of 3 and then shifted up 4 units, what is the new equation g(x)?
g(x) = √(3x) + 4