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# Calculus Functions Study Guide (page 2)

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Updated on Oct 1, 2011

### Parentheses Hint

It is true that in algebra, everyone is taught "parentheses mean multiplication." This means that 5(2 + 7) = 5(9) = 45. If x is a variable, then x(2 + 7) = x(9) = 9x. However, if f is the name of a function, then f(2 + 7) = f(9) = the number to which f takes 9. The expression f(x) is pronounced "f of x" and not "f times x." This can be confusing, so an apology is necessary. Mathematicians use parentheses to mean several different things and expect everyone to know the difference. Sorry!

#### Example 1

Find the value of g(3) if g(x) = x2 + 2.

#### Solution 1

Replace each occurrence of x with 3.

g(3) = 32 + 2

Simplify.

g(3)=9 + 2 = 11

#### Example 2

Find the value of h(–2) if h(t) = t3 –2t2 + 5.

#### Solution 2

Replace each occurrence of t with –2.

h(–2) = (–2)3– 2(–2)2 + 5

Simplify.

h(–2) = – 8 – 2(4) + 5 = – 8 – 8 + 5 = –11

## Plugging Variables into Functions

Variables can be plugged into functions just as easily as numbers can. Often, though, they can't be simplified as much.

#### Example 1

Simplify f(w) if f(x) = √x + 2 x2 + 2.

#### Solution 1

Replace each occurrence of x with w.

f(w) = √w + 2w2 +2

That is all we can say without knowing more about w.

#### Example 2

Simplify g(a + 5) if g(t) = t2 – 3t +1.

#### Solution 2

Replace each occurrence of t with (a + 5).

g(a + 5) = (a + 5)2 – 3(a + 5) + 1

Multiply out (a + 5)2 and –3(a + 5).

g(a + 5) = a2 + 10a + 25 – 3a – 15 + 1

Simplify.

g(a + 5) = a2 + 7a + 11

Simplify .

#### Solution 3

.

Use f(x) = x2 to evaluate f(x + a) and f(x).

Multiply out (x + a)2.

.

Cancel out the x2 and the –x2.

.

Factor out an a.

.

Cancel an a from the top and bottom.

2x + a

## Composition

Now that we can plug anything into functions, we can plug one function into another. This is called composition. The composition of function f with function g is written f g. This means to plug g into f like this: f g(x) = f(g(x))

It may seem that f comes first in f g(x), reading from left to right, but actually, the g is closer to the x. This means that the function g acts on the x first.

#### Example 1

If f(x) = √x + 2x and g(x) = 4x = 7, then what is the composition f g(x)?

#### Solution 1

f g(x) = f(g(x))

Use g(x) = 4x + 7.

f g(x) = f(4x + 7)

Replace each occurrence of x in f with 4x + 7 .

f g(x) = √4x+ 7 + 2(4x+ 7)

Simplify.

f g(x) = √4x+ 7 + 8x + 14

Conversely, to evaluate g f(x) , we compute:

g f(x) = g (f (x))

Use f(x) = √x + 2x.

g f (x) = g(√x + 2 x)

Replace each occurrence of x in g with √x + 2x.

g f (x) = 4(√x + 2x) + 7

Simplify.

g f (x) = 4√x + 8x + 7

Notice that f g(x) and g f (x) are different. This is usually the case.

#### Example 2

If f(x) = x2 + 2x + 1 and g(x) = 5x + 1, then what is f g(x)?

#### Solution 2

f g(x) = f(g(x))

Use g(x) = 5x + 1.

f g(x) = f(5x + 1)

Replace each occurrence of x in f with 5x + 1 .

f g(x) = (5x + 1)2 + 2(5x + 1) + 1

Simplify.

f g(x) = 25x2 + 20x + 4

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