Function & Graph I
What is the set of all possible input values of a function?
Domain
What is the set of all possible output values of a function?
Range
Solve x^2 - 5x + 6 = 0
x = 2 or x = 3
Find the maximum of f(x) = -2x^2 + 8x + 3 using completing the square
Maximum is 11 at x = 2
Amplitude of y = 3 cos x?
3
Which function has an inverse?
One-to-one
Explain why f(x)=x2 does not have an inverse over all real numbers.
It is not one-to-one; each y-value corresponds to two x-values.
What is the discriminant of x^2 - 4x + 5 = 0, and how many real roots exist?
(-4)^2 - 415 = -4 < 0, no real roots
Factor x^3 - 6x^2 + 11x - 6 completely
(x-1)(x-2)(x-3)
Period of y = tan(2x)?
pi/2
If f(x)=3x+2, what is the inverse function?
(x-2)/3
What is fg(x) if f(x)=x^2 and g(x)=x+1
fg(x)=(x+1)^2
Solve x^2 - 5x + 6 > 0
x < 2 or x > 3
Use the remainder theorem to find remainder of f(x) = x^3 + x - 2 divided by x-1
f(1) = 0, remainder = 0
Using sec^2 A = 1 + tan^2 A, find sec A if tan A = 3
sec A = sqrt(10)
How does the graph of y = |f(x)| relate to y = f(x)
All parts of y=f(x) below the x-axis are reflected above the x-axis.
Sketch the relationship between a function and its inverse. What line is the mirror?
Reflection across y=x
Solve |x^2 - 4| = 5
x = 3, -3, 1, -1
Explain the condition for a line to be tangent to a parabola y = ax^2 + bx + c
Quadratic formed by equating line and parabola has discriminant = 0
Sketch y = 2 sin x + 1 and describe vertical shift
Shifted up 1 unit
If f(x)=x3 -2x, explain why it exists
It exists because f(x) is one-to-one;
Solve |2x+5|=7
x= 1 or x=-6
Solve the cubic equation x^3 - 6x^2 + 11x - 6 = 0
x = 1, 2, 3
Determine the range of f(x) = -x^2 + 6x - 5
Maximum at x = 3, f_max = 4, range: (-infinity, 4]
Solve 2 cos^2 x - 1 = 0 for 0 <= x < 2pi
x = pi/4, 3pi/4, 5pi/4, 7pi/4