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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 ) . If we graph both functions on a graphing calculator, we will get the upper and lower semicircles.

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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 that 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

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
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Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
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Kavita
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Thuli
Tomorrow its an revision on factorising and Simplifying...
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Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
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Masum
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Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
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Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
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Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
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Oliver
proof for set theory
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Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
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factoring polynomial
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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