Precalculus Unit 1 - Function
vertical reflection
-f(x) a transformation that reflects a function's graph across the x-axis by multiplying the output by −1
Given a line graph, describe the set of values using interval notation.
1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line. 2. At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot). 3. At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot). 4. Use the union symbol ∪ to combine all intervals into one set if necessary.
Given a function represented by a formula, find the inverse
1. Make sure f is a one-to-one function. 2. Solve for x. 3. Interchange x and y and solve.
Reflective symmetry (function symmetry)
A shape has reflective symmetry if it remains unchanged after a reflection across a line.
reflection
A transformation that "flips" a figure over a mirror or reflection line.
Combining Transformations
Combining vertical transformations: af(x)+k - vertically stretch by a and then vertically shift by k. Combining horizontal transformations: f(bx+h) - horizontally shift by h and then horizontally stretch by 1/b. f(b(x+h)) - horizontally stretch by 1/b and then horizontally shift by h. Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.
Evaluating piecewise functions
Determine which domain interval the input value belongs to and use the associated formula to find the output value. Open circle = value not included Closed circle = value is included
Given the formula for a function, determine the domain and range.
Exclude from the domain any input values that result in division by zero. Exclude from the domain any input values that have nonreal (or undefined) number outputs. Use the valid input values to determine the range of the output values. Look at the function graph and table values to confirm the actual function behavior.
inverse function
For any one-to-one function f(x)=y, a function f^−1(x) is an inverse function of f if f^−1(y)=x. Basically just swap x and y and solve. a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.
Algebra of Functions
For two functions f(x) and g(x) with real number outputs, we define new functions f + g, f - g, fg, and f/g by the relations: (image)
Decreasing function
Graph that falls from left to right.
Increasing function
Graph that rises from left to right. If f(b) > f(a) for any two input values a and b in the given interval where b > a.
Finding the Domain of a Function Defined by an Equation
Identify the input values and input restrictions (values in denominator, values under radicand of an even root)
Vertical Line Test
If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.
We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.
Is it possible for a function to have more than one inverse?
Yes. (f(x)= c −x, where c is a constant, is also equal to its own inverse.)
Is there any function that is equal to its own inverse?
y = x (identity line)
Over what line is a one-to-one inverse function symmetrical to the original function?
TRUE
TRUE OR FALSE: A function can be neither odd nor even.
Finding Domain and Range of Inverse Functions
The outputs of the function f are the inputs to f^−1, so the range of f is also the domain of f^−1. - When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function.
transformation
The process of changing a graph from its parent function by translation, reflection, compression, or stretching.
domain
The set of the first components of the ordered pairs in a relation, or x-coordinates
range
The set of the second components of the ordered pairs in a relation, or y-coordinates
Conventions of interval notation
The smallest number from the interval is written first. The largest number in the interval is written second, following a comma. Parentheses, (or), are used to signify that an endpoint value is not included, called exclusive. Brackets, [or], are used to indicate that an endpoint value is included, called inclusive.
What if a < 0 ?
There will be a combination of a vertical stretch or compression with a vertical reflection
piecewise-defined function
a function that is written using two or more expressions. Can pass the vertical line test if no closed circles overlap
constant function
a linear function of the form f(x)=b that is neither increasing nor decreasing
Conventions of set-builder notation
a method of specifying a set of elements that satisfy a certain condition. It takes the form{x| statement about x} which is read as, "the set of all x such that the statement about x is true." For example,
function
a relation in which each possible input value maps to exactly one output value. "The output is a function of the input"
"toolkit functions"
a set of basic named functions for which we know the graph, formula, and special properties.
relation
a set of ordered pairs
Horizontal Line Test
a test used to determine if the inverse of a relation is a function, aka if the function is one-to-one.
shift
a transformation that will move the whole graph left, right, up, or down.
Local Minimum
a value of the input where a function changes from decreasing to increasing from left to right.
Local Maximum
a value of the input where a function changes from increasing to decreasing from left to right.
Toolkit Functions for Increasing or Decreasing Intervals 3
absolute value formula is an example of a piecewise function
vertical compression
af(x), 0 < a < 1 Reduces all y-values of a function by the same factor between 0 and 1 (again, this does not affect x values)
vertical stretch
af(x), a > 1 A transformation that causes the graph of a function to stretch away from the x-axis when all the y-coordinates are multiplied by a factor a, where a > 1
open interval
an interval that does not include its end points
Composition of functions
combining two functions so that the output of one function is the input of another (making a composite function). (f∘g)(x) = f( g(x)) (f composed with g at x, "f of g of x")
y = f(x)
definition of a function named f.
one-to-one functions
each element of the domain pairs to exactly one unique element of the range. There are no repeated x- or y-values.
horizontal reflection
f(-x) a transformation that reflects a function's graph across the y-axis by multiplying the input by −1
odd function
f(-x)= -f(x) x,y -x,-y, symmetric about the origin
even function
f(-x)= f(x) x,y -x,y symmetric about the y axis
horizontal stretch
f(bx) 0<b<1, graph is stretched by 1/b the stretching of the graph away from the x-axis
horizontal compression
f(bx) b > 1, graph is compressed by 1/b the squeezing of the graph towards the y-axis if b < 0, there will be stretch/compression with a horizontal reflection
vertical shift
f(x)+k k is positive - graph moves up k is negative - graph moves down
horizontal shift
f(x-h) h is positive - graph moves right h is negative - graph moves left (Be sure to pay attention to the negative sign in the new function!)
Evaluating functions
finding 'f(x)' at some specific value 'x'
Average Rate of Change
the change between two output values (or function values) dividied by the difference between the two corresponding input values (x-values) Δy/Δx - do not forget to make the y1 term negative
independent variable
the input values found in the domain
dependent variables
the output values found in the range
