Math 1414: Functions Quiz Review
Sum function: Difference function: Product function: Quotient function:
(f+g)=f(x)+g(x) (f-g)=f(x)-g(x) (f·g)=f(x)·g(x) (f/g)=f(x)/g(x), g(x)≠0
Properties of f(x)=|x|
- domain all real numbers - range y≥0 - x and y intercept: (0,0) - even - decreasing (-∞,0] - increasing [0,∞) - absolute minimum of 0 at x=0
Properties of f(x)=√x
- domain and range are set of nonnegative intergers - x and y intercept: (0,0) - neither odd or even - increasing, [0,∞) - absolute minimum of 0 at x=0
Properties of f(x)=3√x (cube root)
- domain and range are the set of all real numbers - x and y intercept: (0,0) - odd - increasing (-∞,∞) - no local extrema
How can relations be expressed?
- verbally - numerically using a table or set of ordered pairs - graphically by plotting points mapping by drawing an arrow from element in domain to the corresponding element in the range
What is the Average Rate of Change?
=Δx/Δy=f(b)-f(a)/b-a, a≠b
What is a relation?
A correspondence between two sets; a set x, called domain, and a set y called the range. In, a relation each element from the domain corresponds to at least one element of the range.
What is a local maximum?
A function has a local maximum at c if there is an open interval in l containing c so that f(c)≥f(x) for all x in this open interval. The number of f(x) is called a local maximum value of f.
What is a local minimum?
A function has a local minimum at c if there is an open interval in l containing c so that f(c)≤f(x) for all x in this open interval. The number of f(x) is called a local minimum value of f.
What is a vertical line test?
A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.
What is a function?
From x into y is a relation that associates with each element of x to exactly one element of y.
What is the extreme value theorem?
If a function is continuous on a closed interval [a,b], then f has both an absolute minimum and an absoulte maximum [a,b].
What is an absolute maximum?
If there is a number u in l for which f(u)≥f(x) for all x in l, then f has an absolute maximum at u, and the number f(u) is the absolute maximum of f on l.
What is an absolute minimum?
If there is a number v in l for which f(v)≤f(x) for all x in l, then f has an absolute minimum at v, and the number f(v) is the absolute minimum of f on l.
How do you determine an even or odd function?
In an equation if f(-x)=f(x) the function is considered even, and if the f(-x)=-f(x) the function is considered odd. On a graph a function is even if it has symmetry with respect to the y-axis, and odd if it has symmetry with respect to the origin.
What is the slope of a secant line?
The average rate of change of a function from a to b equals the slope of the secant line containing the two points (a,f(a)) and (b,f(b)) on its graph.
What is the difference quotient?
The difference quotient of a function f at x is given by f(x+h)-f(x)/h, and h≠0.
Key Functions
constant Identity square cube square root cube root reciprocal absolute value greatest integer
constant Identity square cube square root cube root reciprocal absolute value greatest integer
f(x)=b f(x)=x f(x)=x^2 f(x)=x^3 f(x)=3√x f(x)=1/x f(x)=|x| f(x)=int(x), greatest integer less than or equal to x
When is a function constant?
on an interval l if, for all choices of l, the values of f(x) are equal
When is a function increasing?
on an interval l if, for any choice of x, and x2 in l, with x1<x2. then f(x1)<f(x2)
When is a function decreasing?
on an interval l if, for any choice of x, x1<x2, then f(x1)>f(x2)
Function notation: implicit form: explicit form:
y=f(x) x and y y=f(x)