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# 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