Suppose the value of a digital camera years after it was bought is represented by the function .
- Can you determine the value of and explain what the solution means in the context of this problem?
- Can you determine the value of when and explain what this situation represents?
- What was the original cost of the digital camera?
Watch This
Khan Academy Functions as Graphs
Guidance
A function machine shows how a function responds to an input. If I triple the input and subtract one, the machine will convert into . So, for example, if the function is named , and 3 is fed into the machine, comes out.
When naming a function the symbol is often used. The symbol is pronounced as “ of .” This means that the equation is a function that is written in terms of the variable . An example of such a function is . Functions can also be written using a letter other than and a variable other than . For example, and . 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
When a function is represented as an equation, an ordered pair can be determined by evaluating various values of the assigned variable. Suppose . To calculate substitute:
Graphically, if , this means that the point (4, 8) is a point on the graph of the line.
Example A
If find.
a)
b)
c)
Solution:
To determine the value of the function for the assigned values of the variable, substitute the values into the function.
Example B
Functions can also be represented as mapping rules. If find the following in simplest form:
a)
b)
c)
Solution:
a)
b)
c)
Example C
Let .
a) Evaluate
- i)
- ii)
- iii)
b) Find a value of ‘ ’ where does not exist.
c) Find in simplest form
d) Find ‘ ’ if
Solution:
a)
b) The function will not exist if the denominator equals zero because division by zero is undefined.
Therefore, if , then does not exist.
c)
d)
Concept Problem Revisited
The value of a digital camera years after it was bought is represented by the function
- Determine the value of and explain what the solution means to this problem.
- Determine the value of when and explain what this situation 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 that shows a relationship where there is only one output for every input. In other words, for every value of , there is only one value for .
Guided Practice
1. If find:
- i)
- ii)
2. If find ‘ ’ if
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, , that the truck rolls is represented by the function .
- 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.
Answers:
1.
- i)
- ii)
2.
3.
- i)
- After 9 seconds, the truck will roll 40.5 feet.
- ii)
- The truck will hit the tree in approximately 35 seconds.
Practice
If , find expressions for the following:
If , determine the value of ‘ ’ when:
The value of a Bobby Orr rookie card years after its purchase is .
- Determine the value of and explain what the solution means.
- Determine the value of when and explain what this situation represents.
- Determine the original price of the card.
Let .
- When is undefined?
- For what value of does ?