Welcome to an introduction to function notation.
If a relation is a function, then we say that the output
is a function of the input.
We can communicate this information
using what's called function notation,
which is written here.
Again, a special notation used
to let us know that the output is a function of the input,
so using this notation, notice how the input
always goes inside the parenthesis.
F is the name of our function.
We don't have to call the function f,
though it is common to call a function f,
and on the right, we have the output,
also known as the function value.
As an additional example, if y is a function of x,
where x is the input, and y is the output,
we can write f of x equals y.
Using this notation, let's us know y is a function of x.
X is the input, y is the output,
and the function name is f.
Let's look at an example.
The function v of m represents the value of an investment
in thousands of dollars after m months.
Explain the meaning of v of 36 equals 17.4
So, looking at our function notation here,
notice the input is 36, where the input is m,
which represents the number of months.
The output is 17.4.
The output is v of m,
which represents the value of an investment
in thousands of dollars.
So, if v of 36 equals 17.4, this tells us
that after 36 months, an investment is worth.
Need to be careful here, 17.4 is in thousands of dollars,
so to convert to dollars, we have to multiply by 1,000,
which would give us 17,400,
so this function notation tells us
after 36 months, an investment is worth $17,400.
In example two, we're asked to convert back and forth
between ordered pairs and function notation.
We know for an ordered pair, the first value is the input,
and the second value is the output,
and now we know for function notation
f of the input equals the output,
so notice how the first value here in the ordered pair
would be the same as this value here in function notation,
and this value here of the output
would be the same value here in function notation.
So, if the ordered pair was 2,3,
in function notation, this would be f of two equals three.
Again, we have the input here and here.
We have the output here and here.
So, here we're given the ordered pair
and asked to write function notation,
so because the input is negative four,
and the output is six, in function notation we would have
f of negative four equals six.
Here we have the input, and here we have the output.
And here, we're given the function notation
f of five equals negative one,
so the input is five,
which would be the first value in the ordered pair,
and the output is negative one,
which would be the second value in the ordered pair.
In example three, we have several ordered pairs.
We want to complete the following function notation.
So for f of five, we know that the input is five.
For the ordered pairs, five is the first value,
so notice how when we have an input of five,
the output is seven, and, therefore,
f of five equals seven.
And, next, we have f of some input equals an output of zero,
so here we're given the output of zero,
so looking at the ordered pairs,
we're now looking for the second coordinate,
or the second value of zero,
which notice occurs here.
Notice, when the output is zero, the input must be eight,
which means f of eight equals zero.
Example four, the function b of t is defined
by the table below.
So for b of t, notice that t would be the input,
so this first row gives us the inputs.
The second row gives us the outputs or the function values,
so we first have to find b of 12, so 12 is the input,
or the value of t.
Looking at our table, here's the t value of 12,
or the input of 12.
Notice how when the input is 12, the output,
or function value, would be 50,
and therefore b of 12 equals 50.
Next, we're given that b of t equals 18,
and have to find t the input.
So, here we're given that the output,
or function value, is 18.
Notice how here's a function value of 18,
where b of t equals 18.
This occurs when the input, or t, is equal to 31.
So, if b of t equals 18, t equals 31.
Now, let's look at an example graphically.
Here, we're given the graph of g of x,
and we're going to complete the following.
First we have g of two equals some value.
Well, if we're given g of two,
that means we're given the input of two.
We know we find the inputs along the horizontal axis,
so we'll locate two on the horizontal axis,
which would be here, and notice when the input is two,
the corresponding point on the function
would be this point here.
Notice how the output would be this value here
of positive three,
so the coordinates of this point are 2,3,
which means when the input is two,
the output is three, and, therefore,
g of two equals three,
and we know the ordered pair is 2,3.
Next we have g of zero, so now they inputted zero.
You're going to find the output when the input is zero,
so we locate zero on the horizontal axis, which is here.
The corresponding point in the function would be
this point here.
Notice how the coordinates of this point are 0,4,
so when the input is zero, the output is four,
and therefore g of zero equals four,
and the ordered pair is 0,4.
And now for these last two,
notice how we're given the outputs
and asked to find the inputs.
We locate the outputs on the vertical axis,
so if we know the function value is two,
we locate two on the vertical axis,
which would be here.
The point in our function that has this output
would be this point,
which, notice, has an input of positive four.
The ordered pair here would be 4,2,
where when the input is four, the output is two,
and therefore g of four is equal to two.
Our ordered pair is 4,2.
And for our last example,
we're given the function value at one,
and asked to find the input,
so we first locate one on the vertical axis
where we find the outputs,
which would be here.
The point on the function with an output of one is here,
and notice how the input is six,
so the ordered pair is 6,1,
so we know g of six equals one,
and the ordered pair is 6,1.
Okay, I hope you found this helpful.