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# 3.3: Function Notation

Difficulty Level: Advanced Created by: CK-12
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Practice Function Notation
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Suppose the value V\begin{align*}V\end{align*} of a digital camera t\begin{align*}t\end{align*} years after it was bought is represented by the function V(t)=87550t\begin{align*}V(t) = 875 - 50t\end{align*}.

• Can you determine the value of V(4)\begin{align*}V(4)\end{align*} and explain what the solution means in the context of this problem?
• Can you determine the value of t\begin{align*}t\end{align*} then V(t)=525\begin{align*}V(t) = 525\end{align*} and explain what this represents?
• What was the original cost of the digital camera?

### Guidance

A function machine shows how a function responds to an input. If I triple the input and subtract one, the machine will convert x\begin{align*}x\end{align*} into 3x1\begin{align*}3x - 1\end{align*}. So, for example, if the function is named f\begin{align*}f\end{align*}, and 3 is fed into the machine, 3(3)1=8\begin{align*}3(3) - 1 = 8\end{align*} comes out.

When naming a function the symbol f(x)\begin{align*}f(x)\end{align*} is often used. The symbol f(x)\begin{align*}f(x)\end{align*} is pronounced as “f\begin{align*}f\end{align*} of x\begin{align*}x\end{align*}.” This means that the equation is a function that is written in terms of the variable x\begin{align*}x\end{align*}. An example of such a function is f(x)=3x+4\begin{align*}f(x) = 3x+4\end{align*}. Functions can also be written using a letter other than f\begin{align*}f\end{align*} and a variable other than x\begin{align*}x\end{align*}. For example, v(t)=2t25\begin{align*}v(t) = 2t^2 - 5\end{align*} and d(h)=4h3\begin{align*}d(h) = 4h-3\end{align*}. In addition to representing a function as an equation, you can also represent a function:

• As a graph
• As ordered pairs
• As a table of values
• As an arrow or mapping diagram
• As mapping notation

When a function is represented as an equation, an ordered pair can be determined by evaluating various values of the assigned variable. Suppose f(x)=3x4\begin{align*}f(x)=3x-4\end{align*}, then to calculate f(4),\begin{align*}f(4),\end{align*} substitute:

f(4)f(4)f(4)=3(4)4=124=8

Graphically, if f(4)=8\begin{align*}f(4) = 8\end{align*}, this means that the point (4, 8) is a point on the graph of the line.

#### Example A

If f(x)=x2+2x+5\begin{align*}f(x) = x^2 + 2x +5\end{align*} find.

a) f(2)\begin{align*}f(2)\end{align*}

b) f(7)\begin{align*}f(-7)\end{align*}

c) f(1.4)\begin{align*}f(1.4)\end{align*}

Solution:

To determine the value of the function for the assigned values of the variable, substitute the values into the function.

f(x)=x2+2x+5  f(2)=(2)2+2(2)+5f(2)=4+4+5f(2)=13f(x)=x2+2x+5     f(7)=(7)2+2(7)+5 f(7)=4914+5f(7)=40f(x)=x2+2x+5     f(1.4)=(1.4)2+2(1.4)+5 f(1.4)=1.96+2.8+5f(1.4)=9.76

#### Example B

Functions can also be represented as mapping rules. If g(x)52x\begin{align*}g(x)\rightarrow 5-2x\end{align*} find the following in simplest form:

a) g(y)\begin{align*}g(y)\end{align*}

b) g(y3)\begin{align*}g(y-3)\end{align*}

c) g(2y)\begin{align*}g(2y)\end{align*}

Solution:

g(x)52x  g(y)52(y)g(y)52y  g(x)52x g(2y)52(2y)g(2y)54yg(x)52x    g(y3)52(y3) g(y3)52y+6g(y3)112yApply the distributive propertyCombine like terms

#### Example C

If P(a)=2a3a+2\begin{align*}P(a)=\frac{2a-3}{a+2}\end{align*}:

a) Evaluate

i) P(0)\begin{align*}P(0)\end{align*}
ii) P(1)\begin{align*}P(1)\end{align*}
iii) \begin{align*}P \left ( -\frac{1}{2} \right )\end{align*}

b) Find a value of ‘\begin{align*}a\end{align*}’ where \begin{align*}P(a)\end{align*} does not exist.

c) Find \begin{align*}P(a-2)\end{align*} in simplest form

d) Find ‘\begin{align*}a\end{align*}’ if \begin{align*}P(a)=-5\end{align*}

Solution:

a)

b) \begin{align*}P(a) = \frac{2a-3}{a+2}\end{align*} The function will not exist if the denominator equals zero.

Therefore, if \begin{align*}a=-2\end{align*}, then \begin{align*}P(a)=\frac{2a-3}{a+2}\end{align*} does not exist.

c)

d)

#### Concept Problem Revisited

The value \begin{align*}V\end{align*} of a digital camera \begin{align*}t\end{align*} years after it was bought is represented by the function \begin{align*}V(t) = 875 - 50t\end{align*}

• Determine the value of \begin{align*}V(4)\end{align*} and explain what the solution mean to this problem.
• Determine the value of \begin{align*}t\end{align*} then \begin{align*}V(t) = 525\end{align*} and explain what this represents.
• What was the original cost of the digital camera?

Solution:

• The camera is valued at $675, 4 years after it was purchased. • The digital camera has a value of$525, 7 years after it was purchased.

• The original cost of the camera was \$875.

### Vocabulary

Function
A function is a set of ordered pairs \begin{align*}(x, y)\end{align*} that shows a relationship where there is only one output for every input. In other words, for every value of \begin{align*}x\end{align*}, there is only one value for \begin{align*}y\end{align*}.

### Guided Practice

1. If \begin{align*}f(x)=3x^2-4x+6\end{align*} find:

i) \begin{align*}f(-3)\end{align*}
ii) \begin{align*}f(a-2)\end{align*}

2. If \begin{align*}f(m)=\frac{m+3}{2m-5}\end{align*} find ‘\begin{align*}m\end{align*}’ if \begin{align*}f(m) = \frac{12}{13}\end{align*}

3. The emergency brake cable in a truck parked on a steep hill breaks and the truck rolls down the hill. The distance in feet, \begin{align*}d\end{align*}, which the truck rolls, is represented by the function \begin{align*}d = f(t)=0.5t^2\end{align*}.

i) How far will the truck roll after 9 seconds?
ii) How long will it take the truck to hit a tree which is at the bottom of the hill 600 feet away? Round your answer to the nearest second.

1. \begin{align*}f(x) = 3x^2 - 4x + 6\end{align*}

i)
ii)

2.

3. \begin{align*}d=f(t)=0.5^2\end{align*}

i)
After 9 seconds, the truck will roll 40.5 feet.
ii)
The truck will hit the tree in approximately 35 seconds.

### Practice

If \begin{align*}g(x)=4x^2-3x+2\end{align*}, find expressions for the following:

1. \begin{align*}g(a)\end{align*}
2. \begin{align*}g(a-1)\end{align*}
3. \begin{align*}g(a+2)\end{align*}
4. \begin{align*}g(2a)\end{align*}
5. \begin{align*}g(-a)\end{align*}

If \begin{align*}f(y) = 5y-3\end{align*}, determine the value of ‘\begin{align*}y\end{align*}’ when:

1. \begin{align*}f(y) = 7\end{align*}
2. \begin{align*}f(y) = -1\end{align*}
3. \begin{align*}f(y) = -3\end{align*}
4. \begin{align*}f(y) = 6\end{align*}
5. \begin{align*}f(y) = -8\end{align*}

The value of a Bobby Orr rookie card \begin{align*}n\end{align*} years after its purchase is \begin{align*}V(n)=520+28n\end{align*}.

1. Determine the value of \begin{align*}V(6)\end{align*} and explain what the solution means.
2. Determine the value of \begin{align*}n\end{align*} when \begin{align*}V(n)=744\end{align*} and explain what this represents.
3. Determine the original price of the card.

### Vocabulary Language: English

Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

Jan 16, 2013

Feb 26, 2015