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Evaluating functions

Given the function h ( p ) = p 2 + 2 p , evaluate h ( 4 ) .

To evaluate h ( 4 ) , we substitute the value 4 for the input variable p in the given function.

  h ( p ) = p 2 + 2 p   h ( 4 ) = ( 4 ) 2 + 2 ( 4 )          = 16 + 8          = 24

Therefore, for an input of 4, we have an output of 24.

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Given the function g ( m ) = m 4 , evaluate g ( 5 ) .

g ( 5 ) = 1

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Solving functions

Given the function h ( p ) = p 2 + 2 p , solve for h ( p ) = 3.

                  h ( p ) = 3             p 2 + 2 p = 3 Substitute the original function  h ( p ) = p 2 + 2 p .       p 2 + 2 p 3 = 0 Subtract 3 from each side .   ( p + 3 )( p 1 ) = 0 Factor .

If ( p + 3 ) ( p 1 ) = 0 , either ( p + 3 ) = 0 or ( p 1 ) = 0 (or both of them equal 0). We will set each factor equal to 0 and solve for p in each case.

( p + 3 ) = 0 , p = 3 ( p 1 ) = 0 , p = 1

This gives us two solutions. The output h ( p ) = 3 when the input is either p = 1 or p = 3. We can also verify by graphing as in [link] . The graph verifies that h ( 1 ) = h ( 3 ) = 3 and h ( 4 ) = 24.

Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).
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Given the function g ( m ) = m 4 , solve g ( m ) = 2.

m = 8

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Evaluating functions expressed in formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation 2 n + 6 p = 12 expresses a functional relationship between n and p . We can rewrite it to decide if p is a function of n .

Given a function in equation form, write its algebraic formula.

  1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
  2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Finding an equation of a function

Express the relationship 2 n + 6 p = 12 as a function p = f ( n ) , if possible.

To express the relationship in this form, we need to be able to write the relationship where p is a function of n , which means writing it as p = [ expression involving n ] .

2 n + 6 p = 12 6 p = 12 2 n Subtract  2 n  from both sides . p = 12 2 n 6 Divide both sides by 6 and simplify . p = 12 6 2 n 6 p = 2 1 3 n

Therefore, p as a function of n is written as

p = f ( n ) = 2 1 3 n
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Expressing the equation of a circle as a function

Does the equation x 2 + y 2 = 1 represent a function with x as input and y as output? If so, express the relationship as a function y = f ( x ) .

First we subtract x 2 from both sides.

y 2 = 1 x 2

We now try to solve for y in this equation.

y = ± 1 x 2    = + 1 x 2  and  1 x 2

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function y = f ( x ) .

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If x 8 y 3 = 0 , express y as a function of x .

y = f ( x ) = x 3 2

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Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation x = y + 2 y , if we want to express y as a function of x , there is no simple algebraic formula involving only x that equals y . However, each x does determine a unique value for y , and there are mathematical procedures by which y can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for y as a function of x , even though the formula cannot be written explicitly.

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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cm
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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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Can you compute that for me. Ty
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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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