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Section 1. ● Real numbers
Set Builder notation
Interval notation
Functions a function is the set of all possible points y that are mapped to a single point x. If when x=5 y=4,5 then it is not a function because when graphed it will not pass the vertical line test.
x is the independent variable because it represents values in the domain
y is the dependent variable because it represents values in the range
Implied domain= all of the possible real numbers on the domain that keep the equation real
piecewise defined function= a function defined by multiple equations
Section 1.
Zero the x intercepts Roots the solution to the equation when x=
Line Symmetry when a graph is mirror around the vertex Point Symmetry When a graph can be rotated 180 degrees about the axis and still be the same
Even functions Functions symmetric about the y axis
Odd functions Functions that are symmetric about the x axis
➔ Quadratic Function: f(x)= x^2 is a parent function.
➔ Cubic Function: f(x)=x^3 is the parent function.
➔ Square Root Function: f(x)= (^) √ x
➔ Reciprocal Function: f(x)=1/x
➔ Transformations: the changing of a parent graph that may affect the appearance of the graph but is derived from the parent graphs formula. ➔ Translations: the shifting of a parent graph. Vertical translations shift the graph up or down, while horizontal transformations shift the graph left or right. ➔ Reflections: Mirror image of the graph over a certain axis. ➔ Dilation: The expansion or compression of a graph vertically or horizontally.
f ( x ) = a • ( x − h ) + k a: control dilation of the graph h: horizontal shift k: vertical shift Sign on a determines whether the graph is reflected over the x axis. Sign on x determines whether the graph is reflected over the y axis.
1 6: Function Operations and Composition of Functions
In function operations, they give you a formula (shown above) for two functions and tell you to
find the sum, product, difference, or quotient for a new, combined function. Function operation is simply adding, multiplying, subtracting, or dividing two formulas.
Ex: Given f(x) = x + 3 and g(x) = x^2 + 5, find (f + g)(x).
(f + g)(x) = ((x + 3) + ( x^2 + 5)) = (x + 3 + x^2 + 5) = ( x^2 + x + 8)
Answer: x^2 + x + 8
If asked to find (f + g)(x), (f – g)(x), (f×g)(x), and (f / g)(x) with x being a specific value (such as x = 2), simply find the value of the function at given value x into the equations f(x) and g(x) and plug those answers into (f + g)(x), (f – g)(x), (f×g)(x), or (f / g)(x).
Ex: Given f(x) = x + 3 and g(x) = x^2 + 5, find (f×g)(2)
f(2) = 5 g(2) = 1 (f×g)(2) = f(2) x g(2) = 5 x 1 = 5
Answer: 5
In a composition, we are trying to find the formula that result from plugging the formula f(x) and g(x).
Ex: Given f(x) = x + 3 and g(x) = – x^2 + 5 find ( f o g )(x)
Answer: (f / g)(x) =
- ( f o g )(x) = f(g(x)) = f(–5 x^2 + 3x 16) = (^) 4(–5 x 2 + 3 x^1 − 16) + 9 = (^) −20 x 2 + 12^1 x − 64 + 9 = (^) −20 x 2 + 12^1 x − 55
Answer: ( f o g )(x) = (^) −20 x 2 + 12^1 x − 55
Section 1.
Inverse relationships: Any function that is flipped over the y=x line
Inverse functions These functions are inverse to each other The (X)’s and (Y)’s are switched
Horizontal Line Test
(If the horizontal line hits more than one point than it is not a function) Functions that pass the Horizontal line test are said to be called “One to one”
Finding an Inverse Functions: When finding the inverse of an equation, you switch the x’s and y’s and then solve for Y
