Help
Options
Didn't know it?
click below
click below
Knew it?
click below
click below
Don't Know
Remaining cards (0)
Know
retry
shuffle
restart
0:00
Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.
Normal Size Small Size show me how
Normal Size Small Size show me how
Math1050 CH03
Functions and their graphs
| Term | Definition |
|---|---|
| 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
Popular Math sets