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Math1050 CH03

Functions and their graphs

TermDefinition
When the value of one variable relates to a second variable. i.e., a correspondence between 2 sets relation
If x and y are 2 elements on these sets and a relationship exists the we say x ___ to y or that y ___ x. corresponds, depends on
for the equation y=3x-1 we say x serves as the ___ to the relation and y is the ___ input, output
For each element x in the domain, its corresponding y value is called the ___ of x. image
for y=f(x), x is called the ___ variable, or the ___. independent, argument
for y=f(x), y is called the ___ variable dependent
f(x+h)-f(x) ----------- is called the ___ h difference quotient
If a function is defined by an equation in x & y (e.g., 3x+y=5), we say the function f is given ___. implicitly (it is implied)
If a function is defined by an equation in x & y (e.g., 3x+y=5), and we can solve for y in terms of x (y=f(x)=-3x+5) we write y=f(x) and say it is given ___. explicitly
If a domain is not specified, the domain is understood to be ____ the largest set of real numbers for which the value f(x) is a real number
(f+g)(x) = ___. The domain of f+g consists of the numbers x that are in the domains of both f & g. f(x) + g(x)
(f-g)(x) = ___. The domain of f-g consists of the numbers x that are in the domains of both f & g. f(x) - g(x)
(f*g)(x) = ___. The domain of f*g consists of the numbers x that are in the domains of both f & g. f(x) * g(x)
(f/g)(x) = ___. The domain of f/g consists of the numbers x that are in the domains of both f & g. Where g(x) does not = 0. (f(x))/(g(x)), where g(x) does not = 0
Vertical-line test for a function states that if it is a function, a vertical line will ___. only ever pass through one point
The words odd and even regarding functions describe the ___ of the graph of the fxn. symmetry
A function is ___ if, for every number x in its domain, the number -x is also in the domain AND f(-x)=f(x) even
A graph is even, if and only if, whenever the point (x,y) is on the graph, the point (___,___) is also on the graph. (-x,y)
A function is ___ if, for every number x in its domain, the number -x is also in the domain AND f(-x)=-f(x) odd
A graph is odd, if and only if, whenever the point (x,y) is on the graph, the point (___,___) is also on the graph. (-x,-y)
Even functions are symmetric about the ___. y-axis
Odd functions are symmetric about the ___. origin
To determine odd/even algebraically replace ___ with ___. x, -x
Is f(x)=x2−5 odd or even? f(−x)=(−x)2−5=x2−5=f(x) even
Is g(x)=x3−1 odd or even? g(−x)=(−x)3−1=−x3−1 neither
Is h(x)=5x3−x odd or even? h(−x)=5(−x)3−(−x)=−5x3+x=−(5x3−x)=−h(x) odd
Is F(x)=|x| odd or even? F(−x)=|−x|=|−1|⋅|x|=|x|=F(x) even
A function f is ___ on an open interval I if, for any choice of x1 and x2 in I, with x1<x2, we have f(x1)<f(x2). increasing
A function f is ___ on an open interval I if, for any choice of x1 and x2 in I, with x1<x2, we have f(x1)>f(x2). decreasing
A function f is ___ on an open interval I if, for any choice of x in I, the values f(x) are equal. constant
A function of f has a local ___ at c if there is an open interval I containing c so that for all x in I, f(x)≤f(c). We call f(c) a local ___ value of f. maximum
A function has a local ___ at c if there is an open interval I containing c so that, for all x in I, f(x)≥f(c). We call f(c) a local ___ value of f. minimum
If there is a number u in I for which f(x)≤f(u) for all x in I, then f(u) is the ___ of f on I and we say the ___ of f occurs at u . absolute maximum
If there is a number v in I for which f(x)≥f(υ) for all x in I, then f(υ) is the ___ of f on I and we say the ___ of f occurs at v. absolute minimum
The absolute maximum and absolute minimum of a function f are sometimes called the ___ of f on I. extreme values
To find the average rate of change of a function between any two points on its graph, calculate the ___ of the line containing the two points. slope
Average rate of change = Δy/___ = (f(b)−___)/b−___ Where a≠b Δx, f(a), a
The average rate of change of a function from a to b equals the slope of the ___ containing the two points (a,f(a)) and (b,f(b)) on its graph. secant line
The secant line is the ___ of a right triangle. hypotenuse or slope
Properties of f(x)=√x 1. The domain and the range are the set of all ___. nonnegative real numbers
Properties of f(x)=√x 2. The x-intercept of the graph of f(x)=√x is ___. The y-intercept of the graph of f(x)=√x of is ___. 0, 0
Properties of f(x)=√x 3. The function is even or odd? neither
Properties of f(x)=√x 4. The function is ___ on the interval (0, ∞). increasing
Properties of f(x)=√x 5. The function has an absolute minimum of ___ at x=___. 0
Properties of f(x)=^3√x (cube root) 1. The domain and the range are the set of ___. all real numbers
Properties of f(x)=^3√x (cube root) 2. The x-intercept of the graph of f(x)=^3√x is ___. The y-intercept of the graph of f(x)=^3√x is also ___. 0, 0
Properties of f(x)=^3√x (cube root) 3. The graph is symmetric with respect to the ___ so the function is ___. origin, odd
Properties of f(x)=^3√x (cube root) 4. The function is ___ on the interval (−∞, ∞). increasing
Properties of f(x)=^3√x (cube root) 5. The function has ___ local minima or any local maxima. no
Properties of f(x)=|x| 1. The domain is the set of ___. The range of f is ___. all real numbers, {y|y≥0}(all positive numbers)
Properties of f(x)=|x| 2. The x-intercept of the graph of f(x)=|x| is ___. The y-intercept of the graph of f(x)=|x| is ___. 0, 0
Properties of f(x)=|x| 3. The graph is symmetric with respect to the ___ so the function is ___. y-axis, even
Properties of f(x)=|x| 4. The function is ___ on the interval (−∞, 0). It is ___ on the interval (0, ∞) decreasing, increasing
Properties of f(x)=|x| 5. The function has an absolute minimum of ___ at x=___. 0, 0
f(x)=b where b is a real number is the ___ fxn. constant
f(x)=x is the ___ fxn. Identity
f(x)=x^2 is the ___ fxn. Square
f(x)=x^3 is the ___ fxn. Cube
f(x)=√x is the ___ fxn. Square Root
f(x)=^3√x is the ___ fxn. Cube Root
f(x)=1/x is the ___ fxn. Reciprocal
f(x)=|x| is the ___ fxn. Absolute Value
f(x) = int(x) = greatest integer less than or equal to ___ is the ___ fxn. x, Greatest Integer
When a function is defined by different equations on different parts of its domain, it is called a ___ function. piecewise-defined
If a positive real number k is added to the output of a function y=f(x), the graph of the new function y=f(x)+k is the graph of f shifted ___ k units. vertically up
If a positive real number k is subtracted from the output of a function y=f(x), the graph of the new function y=f(x)−k is the graph of f shifted ___ k units. vertically down
If the argument x of a function f is replaced by x−h, h>0, the graph of the new function y=f(x−h) is the graph of f shifted ___ h units. horizontally right
If the argument x of a function f is replaced by x+h, h>0, the graph of the new function y=f(x+h) is the graph of f shifted ___ h units. horizontally left
If a positive number h is subtracted from x in y=f(x), the graph of the new function y=f(x−h) is the graph of y=f(x) shifted ___ h units. If h is added to x, shift ___ h units. horizontally right, horizontally left
When the right side of a function y=f(x) is multiplied by a (+) number a, the graph of the new function y=af(x) is obtained by multiplying each y-coordinate by a. The new graph is a vertically ___ if 0<a<1 or a vertically ___ if a>1 compressed, stretched
If the argument x of a function y=f(x) is multiplied by a positive number a, the graph of the new function y=f(ax) is obtained by multiplying each x-coordinate of y=f(x) by 1a. A horizontal ___ results if a>1, and a horizontal ___ occurs if 0<a<1. compression, stretch
When the right side of the function y=f(x) is multiplied by −1, the graph of the new function y=−f(x) is the reflection about the ___ of the graph of the function y=f(x). x-axis
When the graph of the function y=f(x) is known, the graph of the new function y=f(−x) is the reflection about the ___ of the graph of the function y=f(x). y-axis
If f(x)=|x| is changed to g(x)=2|x| the new graph is ___. y=2f(x) vertically stretched by a factor of 2 (each y value is doubled)
If f(x)=|x| is changed to g(x)=1/2|x| the new graph is ___. y=1/2f(x) vertically compressed by a factor of 1/2 (each y value is halved)
2f(x) yields a graph that is ___. vertically stretched by a factor of 2 (vertically doubled)
1/2f(x) yields a graph that is ___. vertically compressed by a factor of 2 (vertically halved)
f(2x) yields a graph that is ___. horizontally compressed by a factor of 1/2 (horizontally halved)
f(1/2x) yields a graph that is ___. horizontally stretched by a factor of 2 (horizontally doubled)
Choose: vertical or horizontal. You apply the ___ stretches & compressions to everything in the function, whereas you only apply the ___ stretches & compressions the terms where the independent variable (x) exists. vertical, horizontal
Created by: drjolley