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| 10 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
There are different ways to represent a function — using tables of values, mapping diagrams, graphs, and equations. However, aside from these, one of the most common ways is using function notation.
Function notation is a special way to write functions that explicitly shows that y is a function of x — in other words, that y depends on x. Function notation is symbolically expressed as y=f(x) and read y equals f of x.
Equations that are functions can be written using function notation.
ccc
Equation & & Function Notation [1ex]
y=-5x+4 & & f(x) = -5x+4
Notice that y has been replaced by f(x). In function notation, x represents an element of the domain and f(x) represents the element of the range that corresponds to x. When written in function notation, the expression that describes how to convert an input into an output — the right-hand side expression — is called the function rule.
Besides f, other letters such as g or h can be used to name the function. Similarly, letters other than x can name the independent variable.
To interpret an equation given in function notation, it is necessary to understand what both sides of f(x)=k mean. For example, consider the following equation. f( 3) = 12 Here, f( 3) denotes that the function's input is x= 3 and that 12 is the output corresponding to this input. f( 3) = 12 ⇓ The output offwhenx= 3 is 12. Now, consider a different scenario. Let w(t)=200t be a function that describes the number of words Kevin reads in t minutes. The following statements are true for this function. w(4) and w(t)=900 Here, w(4) is the number of words that Kevin reads in 4 minutes and can be found by evaluating the function for t=4. However, the input is not a particular number in the second statement. In such cases, the statement can be interpreted as a question. w(t)=900 ⇓ For which value oft is the output equal to900? Based on the context, the second statement asks how many minutes it takes Kevin to read 900 words. To find this value of t, the equation w(t)=900 has to be solved for t.
In the same way that an expression can be evaluated at a particular x-value, functions can also be evaluated at a specific input. Furthermore, it is also possible to determine the input that produces a specific output.
For the last few days of the long holiday, Izabella and Emily visited a city they did not know. Before returning home, they went into a souvenir shop where all the souvenirs were priced the same. After picking out some gifts and just before paying, Emily found a $3 gift card in her purse.
The function C(s)=4.5s-3 represents the cost, in dollars, of buying s souvenirs with a $3 gift card.
s= 10
Multiply
Subtract term
C(s)= 69
LHS+3=RHS+3
.LHS /4.5.=.RHS /4.5.
Calculate quotient
Rearrange equation
x= 1
Add and subtract terms
Multiply
Subtract term
V(x)= 21
Distribute 9 and -3
Add and subtract terms
.LHS /12.=.RHS /12.
Calculate quotient
Rearrange equation
Two functions f(x) and g(x) can be combined into a sum, a difference, a product, or a quotient through the following formulas. (f+g)(x) &= f(x)+g(x) (f-g)(x) &= f(x)-g(x) (f* g)(x) &= f(x)* g(x) [0.1cm] (f/g) (x) &= f(x)/g(x), g(x)≠ 0 However, apart from these four operations, two functions can also be combined into a composition.
x= g(x)
f(g(x))= h(x)
g(x)= x-5
Distribute 2
Add terms
Izabella and Emily are shopping for pants. Both girls have a $3 discount coupon for the store. After choosing a pair of pants each, they discovered that the pants are 25 % off the marked price. The girls could not agree which order of discount would save them more money, so the cashier allowed them to apply the discounts in whichever order they wanted.
The function f(x)=x-3 represents the effect of applying only the coupon to the marked price, while g(x)= 34x represents the effect of applying only the store's sale discount to the marked price. In both functions, x represents the marked price.
x= f(x)
g(f(x))= I(x)
f(x)= x-3
x= 31
Subtract term
a/c* b = a* b/c
Calculate quotient
x= g(x)
f(g(x))= E(x)
g(x)= 3/4x
E(x)= 21
LHS+3=RHS+3
LHS * 4=RHS* 4
.LHS /3.=.RHS /3.
Rearrange equation
Marked Price | Amount Paid | |
---|---|---|
Izabella's Pants | $31 | $21 |
Emily's Pants | $32 | $21 |
As shown, both girls paid the same amount of money, but the pair of pants that Emily chose was more expensive. As such, Emily saved more money from the discounts. In fact, if Izabella had applied the discounts as Emily did, she could have paid $20.25 for the same pair of pants. E(31) = 3/4(31)-3 = 20.25 In contrast, if Emily had applied the discounts as Izabella did, she would have paid $21.75. Regardless of who applied the discounts better, both girls got a big discount on their purchase and went home happy.
Distribute 1/3
a * 1/a=1
a * 1=a
a/c* b = a* b/c
Calculate quotient
Add terms
To find the required value, we need to know the explicit function rule of g(x), for which we have to know the value of m. g(x) = mx+1 We will use the fact that g(h(x))=f(g(x)) to find the value of m. Let's start by performing the left-hand side composition, which is equivalent to evaluating g(x) at h(x).
Next, on the right-hand side, we will substitute h(x) for its function rule.
Let's perform the composition f(g(x)) by following the same procedure.
Our next step is to equate the two expressions that we wrote. g(h(x))=f(g(x)) ⇓ 3mx + m + 1 = 3mx - 1 Finally, let's solve this equation for m. Note that the linear terms are equal, which means that they will cancel out.
We found that the value of m is -2. This allows us to write the explicit function rule of g(x). g(x) = -2x + 1 ✓ Finally, we can find the value of g(3)- f(g(3)). From the previous computations, we have that f(g(x))=3mx-1 and since m=-2, we get that f(g(x))=-6x-1. Now we are ready to find the required value.
To determine the function rule that defines f(x), we need to find the values of a and b involved in the general form of f(x). f(x) = ax + b We can find these values by using the given equation. f(f(f(x))) = 64x + 189 The left-hand side of the equation implies that we need to perform the composition of f(x) with itself twice. Let's start by finding f(f(x)). To do so, we substitute f(x) for x in the general function rule.
Next, on the right-hand side, we will substitute ax+b for f(x).
To find f(f(f(x))), let's evaluate f(x) at f(f(x)). f( f(f(x))) = a f(f(x)) + b Now, on the right-hand side, we change f(f(x)) for the expression we found before.
We know that this last expression is equal to 64x + 189. In order for the expressions to be equal, the coefficients of the x-terms and the constant terms must be the same on both sides. Let's highlight them. a^3x + a^2b + ab + b = 64x+189 From this equation, a^3 has to be equal to 64 and a^2b+ab+b has to be equal to 189. The only number that makes a^3 equal to 64 is 4. 4^3 = 64 Therefore, a=4. With this information, we can find b.
Now that we know the values of a and b, we can write the function rule that defines f(x). f(x) = ax+b ⇒ f(x) = 4x + 9