Introduction

A function is a rule that takes a number as input and then outputs another number.

For instance, let's look at the following function:

f(x) = 3x+1

In this example,

  • f is the name of the function,

  • the number x is the input,

  • the number 3x+1 is the output.

If we substitute a number into our function in place of x , we get another number as output.

For example, by substituting x={\color{blue}{2}}, we get

f({\color{blue}{2}}) = 3\cdot {\color{blue}{2}}+1 = 7.

So, the input x={\color{blue}{2}} results in an output of 7.

We can substitute any number we like into this function! For example, if we substitute x={\color{red}{-2}}, we get

f({\color{red}{-2}}) = 3\cdot ({\color{red}{-2}})+1 = -5.

So, the input x={\color{red}{-2}} results in an output of -5.

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Example: Evaluating a Function

If g(x) = (x-4)^2+3(5-x) , calculate g(2).

EXPLANATION

We substitute x=2 into the function definition and then simplify: \eqalign{ g(x) &= (x-4)^2+3(5-x) \\[5pt] g(2) &= (2-4)^2+3(5-2) \\[5pt] &= (-2)^2+3(3) \\[5pt] &= 4 + 9 \\[5pt] &= 13 }

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Practice: Evaluating a Function

Evaluate $g(2)$, where $g(x)=x^2-4x+4.$

a
 $g(2)=4$
b
 $g(2)=8$
c
 $g(2)=-4$
d
 $g(2)=0$
e
 $g(2)=-2$
Practice: Evaluating a Function

If $f(t) = 2(t-3)^2+t-1,$ then $f(3)=$

a
b
c
d
e
Practice: Evaluating a Function

If $g(x)= 2\cdot 3^x + 1,$ then $g(2)=$

a
b
c
d
e
Evaluating Functions of More Than One Variable

Functions with more than one input are called multivariable functions.

For example, the multivariable function f(x,y)=x-y takes in two numbers, x and y , and returns their difference.

For multivariable functions, the order of the inputs matters. For example, if we substitute x={\color{blue}{3}} and y={\color{red}{2}}, we get

f({\color{blue}{3}},{\color{red}{2}}) = {\color{blue}{3}}-{\color{red}{2}}=1,

whereas if we substitute x={\color{red}{2}} and y={\color{blue}{3}}, we get a different output:

f({\color{red}{2}},{\color{blue}{3}})={\color{red}{2}}-{\color{blue}{3}}=-1

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Example: Evaluating a Multivariable Function

Evaluate f(4,-1) if f(x,y)= 3\cdot 2^{x+3y}.

EXPLANATION

We substitute x=4 and y=-1 into the function definition and then simplify:

\begin{align*} f(x,y) &= 3\cdot 2^{x+3y} \\[5pt] f\left( 4,-1 \right) & = 3\cdot 2^{4+3(-1)} \\[5pt] & = 3\cdot 2^{4-3} \\[5pt] & = 3\cdot 2^{1} \\[5pt] &=3\cdot 2\\[5pt] &=6 \end{align*}

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Practice: Evaluating a Multivariable Function

Evaluate $f\left(\dfrac{1}{3}, 2 \right)$, where $f(x,y)= 9x-5y+6.$

a
 $1$
b
 $-8$
c
 $2$
d
 $-2$
e
 $-1$
Practice: Evaluating a Multivariable Function

If $g(a, b)= (a-b)(a^2 - b^2),$ then $g(2, -2)=$

a
b
c
d
e
Practice: Evaluating a Multivariable Function

If $f(x,y)= x\sqrt{x^2+y},$ then $f(2,5)=$

a
b
c
d
e
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