Piecewise Functions

John runs in a triathlon. He rides his bike twelve miles during the first part at a rate of six miles per hour. This is followed by swimming a mile (100 laps) at a rate of fifteen seconds a lap. Finally, John jogs ten miles at a rate of 2.5 miles per hour. Create equations that create a continuous graph for all sixteen miles.

1) Make a graph with miles on the Y axis and hours on the X axis. 2) Have all values in the same units.

Graph the piecewise equations Y = 2x^2 for x>0 and Y = -7x -3 for x<= 0. For what identical values of Y are the absolute value of the corresponding x values identical? Where will this graph intersect the x- and y-axes?

1) Set the two equations equal to each other. 2) Create a quadratic equation. 3) Solve for the x-intercepts of the quadratic equation.

How is this used in real life?

Certain chemicals are flammable (can catch on fire or explode) between certain concentrations. Outside of those ranges, the chemical is safe. f(c) = 3c - 18 for c<12 and f(c) = 3c - 18 for c>22 but f(c) = 2c + 3 for 12<c<22. This means you can do one thing with C when there is a high or low concentration, but you must do something else with it during the middle range.

Continuous functions are those that can be drawn without having to lift your pencil or without any gaps in the lines.

Discontinuous functions are those that have gaps in the lines when grafted.

For what domain will the function f(x) = (16 - x)^.5 result in real solutions.

For what domain will the function f(x) = | x^2 - 7 | be greater than the function g(x) = 6x?

Given the equations A) h(x) = x^2-4 for x<3; and, B) h(x) = (2/3)x - 5 for x>=3, 1) graph the equation; 2) determine the x- and y-intercepts; 3) determine the vertex; 4) determine if the graph has a maximum or minimum point; 5) determine if this graph is continuous; and, 6) determine over what domain the range decreases.

Given a graph that consists of A) a ray extending from the point (-2,4) through the point (-5, -5); B) a ray extending from the point (2,4) through the point (5,-5); and, C) a line connecting (-2, 4) and (2, 4), 1) determine the equations; 2) determine the x- and y-intercepts; 3) determine the vertex if there is one; and, 4) determine the domain over which the range increases.

For the equations Y=-2*|x+1| for x<=1, Y=3 for 1<x<3, and Y=6-2x for x>=3, what are the values of Y for x= 10, 2, 0, -1. Are these equations continuous or disjoint?

Given the equations: A) g(x) = 3x + 12 for x<=-3; B) g(x) = |x| for -3<x<3; and, g(x) = -3x+12 for g(x) >= 3, 1) graph the equation; 2) determine the x- and y-intercepts; 3) determine the vertex; 4) determine if the graph has a maximum or minimum point; 5) determine if this graph is continuous; and, 6) determine over what domain the range is decreasing.

f(x) = x^2 + 2 for X<=1 and f(x) = | 6-x | for X>1 Graph the function over the domain {-5, 10} and display the data in a table showing the X and Y coordinates.

Graph the function g(x) = | x + 3 | for X<= -1 and g(x) = x^2 + 1 for X>1

A company determined their shipping costs as a percentage of purchase bands. Items that cost a maximum of $20 pays $8 shipping. Items that cost a maximum of $40 pays $4 shipping. Items that cost a maximum of $80 pays $2. Items that cost more than $80 have free shipping. Create the equations for shipping relative to the cost of items.

Make a graph for Y = 2*|x| - 4 where x<=-2 and x>=2, and Y = x^2 - 4 where -2<x<2. Determine the X- and Y- intercepts and vertex. Is this a continuous or disjoint graph?

How we write functions

Usually you have seen functions written as Y = number sentence including the variable X. An example of this is Y = 3X + 6

Piecewise functions is a way of breaking-down a function, especially when it comes to graphs, into different parts.

What happens to the function f(x) = 1/(x+1) when X = -1?

Functions can be nested, with one function replacing another function. For example, the function of F of G would be...

f(g(x)) where f(x) = X + 3 and g(x) = 7 - X would result in f(g(x) = (7 - X) + 3 where g(x) was substituted for the X in the equation f(x).

Another way of writing functions with number sentences including the variable X replaces the Y with a "function" notation.

f(x) = 3X + 6 is read "the function F of X equals blah blah blah blah." It also can be written g(x) = 3X + 6, which is read "the function G of X equals blah blah."

Since the function f(x) = 1/(x+1) goes to infinity (1/0) when X = -1, we don't want to include that on the graph or in the results.

f(x) = x + 1 for X<-1 and X>-1 would solve this problem. This means to use that function from negative infinity almost to -1 (say -1.000000001), and again use it following -1 (say -.99999999) to positive infinity.