Chapter 7: Functions Flashcards
Shape of an even degree polynomial >2 degrees
Similar to quadratic, maybe w bumps in middle. + = upward slope, - = downward slope
Shape of an odd degree polynomial with >3 degrees
Similar to cubic but maybe w bumps in middle. + = upward sloping, - = downward
What is a power function
F(x) = x^a, such that a is a real number. a is fixed, x changes
What is a reciprocal function?
F(x) = 1/x or x^-a.
Domain and range of R \ {0}
Graph : hyperbola (vert and horiz asymptotes)
What does the graph of x^(1/n) (or the nth root of x) look like? What is its domain?
Even: domain (0, infinity), graph looks like y = x^(1/2) (square root of x). see notes for image
Odd: domain (R), graph looks like y = x^(1/3) (cube root of x) see notes for image
What are absolute value functions?
f(x) = |x|
X is R
Written as piecewise functions for x and -(x).
Graph is reflected on the x-axis for the (-) portion.
What are rational functions?
f(x) = p(x) / q(x)
Where p and q are polynomials and (unless specified otherwise) the domain of f is R such that q(x) ≠ 0.
The graph may have vertical asymptotes at x for q(x) = 0
What is a non-vertical asymptote?
An asymptote that’s not at the origin (?)
What are exponential functions?
f(x) = a^x, where a is R
When a>1, the graph passes through (0,1) and the x-axis is the horizontal asymptote. The graph rises steeply to the RIGHT of the y-axis.
When a<1, the graph passes through (0,1 and the x-axis is the horizontal asymptote. The graph rises steeply to the LEFT of the y-axis.
What does y = e^x look like?
A>1 exponential function that passes through (0,1)
What are logarithmic functions?
f(x) = log a (x), where a is R and ≠ 1.
Domain: (0, infinity).
Passes through (1,0)
Vertical asymptote as y-axis
A>1 : slopes up to the right
A<1 : slopes down the the right (like a slide)
What domain makes sin(x) a 1:1 function? Consequently, what is the inverse function of sin(x) and what are its domain and range?
Domain: [-pi/2,pi/2]
Inverse: arcsin or sin^-1
Domain of inverse: [-1,1]
Range of inverse: [-pi/2,pi/2]
What domain makes cos(x) a 1:1 function? What is its inverse and its inverse’s domain and range?
Domain: [0,pi]
Inverse: arccos
Inverse domain: [-1,1]
Inverse range: [0,pi]
What is the domain that makes tan(x) a 1:1 function? What is its inverse and its inverse’s domain and range?
Domain: (-pi/2,pi/2)
Inverse: arctan
Inverse domain: R
Inverse range: (-pi/2,pi/2)
When can a limit exist?
If approaching from both sides = approaching the same number. BOTH SIDES MUST BE EQUAL.
Dif number = limit DNE
