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Finding the domain and range using toolkit functions

Find the domain and range of f ( x ) = 2 x 3 x .

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is ( , ) and the range is also ( , ) .

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Finding the domain and range

Find the domain and range of f ( x ) = 2 x + 1 .

We cannot evaluate the function at −1 because division by zero is undefined. The domain is ( , −1 ) ( −1 , ) . Because the function is never zero, we exclude 0 from the range. The range is ( , 0 ) ( 0 , ) .

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Finding the domain and range

Find the domain and range of f ( x ) = 2 x + 4 .

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

x + 4 0  when  x 4

The domain of f ( x ) is [ 4 , ) .

We then find the range. We know that f ( 4 ) = 0 , and the function value increases as x increases without any upper limit. We conclude that the range of f is [ 0 , ) .

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Find the domain and range of f ( x ) = 2 x .

domain: ( , 2 ] ; range: ( , 0 ]

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Graphing piecewise-defined functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function f ( x ) = | x | . With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

f ( x ) = x if x 0

If we input a negative value, the output is the opposite of the input.

f ( x ) = x if x < 0

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function    is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S would be 0.1 S if S $ 10 , 000 and $ 1000 + 0.2 ( S $ 10 , 000 ) if S > $ 10 , 000.

Piecewise function

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

f ( x ) = { formula 1     if  x  is in domain 1 formula 2     if  x  is in domain 2 formula 3     if  x  is in domain 3

In piecewise notation, the absolute value function is

| x | = { x     if   x 0 x   if   x < 0

Given a piecewise function, write the formula and identify the domain for each interval.

  1. Identify the intervals for which different rules apply.
  2. Determine formulas that describe how to calculate an output from an input in each interval.
  3. Use braces and if-statements to write the function.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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