Core 3 - Functions

Question 1

What are the two types of functions? Give appropriate examples. What do these functions produce?

Answer 1

Many - 1 e.g. y=sin(x), y=x^2. Many inputs give one y output

1-1 e.g. y=2x+3 One input gives one y output.

Question 2

What type of relationship cannot be considered a function? Give an example. What do these relationships produce?

Answer 2

1-many relationships produce more than one output for each x input.e.g. x=y^2

Question 3

What is the domain of a function?

Answer 3

The x values for which the function is defined.

Question 4

What is the range of a function?

Answer 4

The y values for the suitable x inputs.

Question 5

How do you find the inverse function of f(x)=X^2 +3

Answer 5

Switch x and y around: x=y^2 +3
Solve for y: y=sqrt(x-3)
Replace y with f^-1(x).

Question 6

For a composite function fg(x), what is the order of substitution? Give the equation of fg(x)
f(x)=3x+4
g(x)=x^2

Answer 6

Substitute g(x) into f(x)

fg(x)=3x^2 +4

Question 7

For composite functions, how do we work out the domain and range?

Answer 7

Find the most limiting domain of the functions used, as this will be the domain of the composite function. Substitute the extreme values into the equation to find the range.

Question 8

When finding the inverse function, what is the relationship between domain and range, and what is the final step after having found the inverse function in terms of y?

Answer 8

The domain and range are swapped, as the inverse function is a reflection in y=x. Domain becomes range, and vice versa.

Remember to change the notation from f(x) to f^-1(x) to indicate the inverse function.

Question 9

When stating the domain and range of a function, which notation do we use in the inequality?

Answer 9

Domain
use x in the inequality

Range
Use f(x) in the inequality to indicate an output

Question 10

What are the rules for translations of functions?

Answer 10

Think what X or Y is being replaced by. Then do the translations one step at a time, using opposite of bidmas.

Question 11

If there is a squared term, when finding an inverse, what must there be?

Answer 11

2 solutions. Remember, if you square root, it’s the +-root

Question 12

When solving inverse functions involving squared terms, what must you do?

Answer 12

Check each solution satisfies the domain of the equation, and explicitly exclude values that don’t match.

Question 13

What is the sequence of transformations that maps y=f(x) onto y=1/3 f(x)+1?

Answer 13

replace y with 3y
Replace y with y-1 hence 3(y-1)

so…

Stretch factor 1/3 in y direction
translation by [0,1]

Question 14

What is the sequence of transformations that maps y=g(x) onto y=-f(x)+5

Answer 14

y replaced by -y

replace y with y-5 hence -(y-5)

Question 15

What is the sequence of transformations that maps y=h(x) into y=h(2x-8)

Answer 15

replace x with x-8
replace x with 2x

hence translation by [+8,0]
stretch factor 1/2 in x direction

Or…

replace x with 2x
replace x with x-4
hence 2(x-4)=2x-8

Question 16

When solving a function, which yields two solutions, what MUST YOU LOOK OUT FOR?

Answer 16

Do both solutions fall within the domain of the function? THEY PROBABLY DO NOT. If it is the inverse of a function, consider the range of original function.

Question 17

When finding inverse function involving sqrt, what must you check?

Answer 17

Whether the positive and negative are in the range of the function

Check this by comparing to domain of original function, e.g. if f(x)

Question 18

What is the format for expressing domain and range?

Answer 18

Domain:
x>p

Range
f(x)