# 3.1 Functions and function notation  (Page 5/21)

 Page 5 / 21

## Evaluating functions

Given the function $\text{\hspace{0.17em}}h\left(p\right)={p}^{2}+2p,\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}h\left(4\right).\text{\hspace{0.17em}}$

To evaluate $\text{\hspace{0.17em}}h\left(4\right),\text{\hspace{0.17em}}$ we substitute the value 4 for the input variable $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ in the given function.

$\begin{array}{ccc}\hfill h\left(p\right)& =& {p}^{2}+2p\hfill \\ \hfill h\left(4\right)& =& {\left(4\right)}^{2}+2\left(4\right)\hfill \\ & =& 16+8\hfill \\ & =& 24\hfill \end{array}$

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

Given the function $\text{\hspace{0.17em}}g\left(m\right)=\sqrt{m-4},\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}g\left(5\right).$

$\text{\hspace{0.17em}}g\left(5\right)=1\text{\hspace{0.17em}}$

## Solving functions

Given the function $\text{\hspace{0.17em}}h\left(p\right)={p}^{2}+2p,\text{\hspace{0.17em}}$ solve for $\text{\hspace{0.17em}}h\left(p\right)=3.$

If $\text{\hspace{0.17em}}\left(p+3\right)\left(p-1\right)=0,\text{\hspace{0.17em}}$ either $\text{\hspace{0.17em}}\left(p+3\right)=0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(p-1\right)=0\text{\hspace{0.17em}}$ (or both of them equal 0). We will set each factor equal to 0 and solve for $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ in each case.

$\begin{array}{cccccc}\hfill \left(p+3\right)& =& 0,\hfill & \hfill \phantom{\rule{0.5em}{0ex}}p& =& -3\hfill \\ \hfill \left(p-1\right)& =& 0,\hfill & \hfill \phantom{\rule{0.5em}{0ex}}p& =& 1\hfill \end{array}$

This gives us two solutions. The output $\text{\hspace{0.17em}}h\left(p\right)=3\text{\hspace{0.17em}}$ when the input is either $\text{\hspace{0.17em}}p=1\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}p=-3.\text{\hspace{0.17em}}$ We can also verify by graphing as in [link] . The graph verifies that $\text{\hspace{0.17em}}h\left(1\right)=h\left(-3\right)=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(4\right)=24.$

Given the function $\text{\hspace{0.17em}}g\left(m\right)=\sqrt{m-4},\text{\hspace{0.17em}}$ solve $\text{\hspace{0.17em}}g\left(m\right)=2.$

$\text{\hspace{0.17em}}m=8\text{\hspace{0.17em}}$

## 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 $\text{\hspace{0.17em}}2n+6p=12\text{\hspace{0.17em}}$ expresses a functional relationship between $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}p.\text{\hspace{0.17em}}$ We can rewrite it to decide if $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a function of $\text{\hspace{0.17em}}n.\text{\hspace{0.17em}}$

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 $\text{\hspace{0.17em}}2n+6p=12\text{\hspace{0.17em}}$ as a function $\text{\hspace{0.17em}}p=f\left(n\right),\text{\hspace{0.17em}}$ if possible.

To express the relationship in this form, we need to be able to write the relationship where $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a function of $\text{\hspace{0.17em}}n,\text{\hspace{0.17em}}$ which means writing it as $\text{\hspace{0.17em}}p=\left[\text{expression}\text{\hspace{0.17em}}\text{involving}\text{\hspace{0.17em}}n\right].$

Therefore, $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is written as

$p=f\left(n\right)=2-\frac{1}{3}n$

## Expressing the equation of a circle as a function

Does the equation $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}=1\text{\hspace{0.17em}}$ represent a function with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as input and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as output? If so, express the relationship as a function $\text{\hspace{0.17em}}y=f\left(x\right).$

First we subtract $\text{\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}}$ from both sides.

${y}^{2}=1-{x}^{2}$

We now try to solve for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in this equation.

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $\text{\hspace{0.17em}}y=f\left(x\right).$ If we graph both functions on a graphing calculator, we will get the upper and lower semicircles.

If $\text{\hspace{0.17em}}x-8{y}^{3}=0,\text{\hspace{0.17em}}$ express $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.$

$y=f\left(x\right)=\frac{\sqrt[3]{x}}{2}$

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 $\text{\hspace{0.17em}}x=y+{2}^{y},\text{\hspace{0.17em}}$ if we want to express $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ there is no simple algebraic formula involving only $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that equals $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ However, each $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does determine a unique value for $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ and there are mathematical procedures by which $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ even though the formula cannot be written explicitly.

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
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?"
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
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
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
what is sine?
what is the standard form of 1
1×10^0
Akugry
Evalute exponential functions
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
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
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
factoring polynomial