Functions Intro

Question 1

What’s a relation?

Answer 1

A set of ordered pairs that maps x values to y values

Question 2

What’s a function?

Answer 2

A relation where each x value has one y value. This means one x value cannot have multiple y values. However, a y-value can have multiple x values.

Question 3

What is the notation for a function?

Answer 3

f(x), where f is the function name, x is the input, and f(x) is the output

Question 4

What is domain and range?

Answer 4

Domain is all possible inputs. Range is all possible outputs.

Question 5

For domain and range, you can use set notation or interval notation. State examples for each type of notation.

Answer 5

Set notation: D: {x | x > 0}
Set notation: D: R
Set notation: D: {1, 2, 3}

Interval notation: [0, 5]
Interval notation: (0, 5)
Interval notation (-5, 0) U (0, 5)

[ ] for closed interval
( ) for open interval
U means union

Question 6

What is a base/parent function?

Answer 6

The simplest form of a function, the core equation from which a whole family of functions is derived by applying transformations

Question 7

What is the base function for a linear, quadratic, and cubic function?

Answer 7

Linear: f(x) = x
Quadratic: f(x) = x^2
Cubic: f(x) = x^3

Question 8

What is the base function for an absolute value function? State its general formula w/ transformations, domain, and range.

Answer 8

Base function: f(x) = |x|
General formula: f(x) = a|b(x - h)| + k
Domain: All real numbers
Range:
y >= k if a > 0
y <= k if a < 0

Question 9

What is the base function for a square root function? State its general formula w/ transformations, domain, and range.

Answer 9

Base function: f(x) = √x
General formula: f(x) = a√(b(x - h)) + k
Domain:
x >= h if b > 0
x <= h if b < 0
Range:
y >= k if a > 0
y <= k if a < 0

Question 10

What is the base function for a reciprocal function? State its general formula w/ transformations, domain, range, vertical asymptote, and horizontal asymptote.

Answer 10

Base function: f(x) = 1/x
General formula: f(x) = a/(b(x - h)) + k
Domain: x is not equal to h
Range: y is not equal to k
V.A: x = h
H.A: y = k

Question 11

What is a vertical and horizontal asymptote?

Answer 11

Vertical asymptotes are vertical lines a function approaches but never touches as the function approaches positive/negative infinity.

Horizontal asymptotes are horizontal lines a function a function approaches but never touches as x goes to positive/negative infinity.

Question 12

What do vertical transformations affect? State them

Answer 12

They affect the y-values.

f(x) + k is a shift up/down by k units

a(fx), where |a| > 1, is a vertical stretch by |a| units
a(fx), where 0 <|a|< 1, is a vertical compression by |a|units.

Question 13

What do horizontal transformations affect? State them

Answer 13

They affect the x-values

f(x-h) is a shift to the right/left by |h| units
f(bx), where |b| > 1, is a horizontal compression by 1/|b| units
f(bx), where 0 < |b| < 1, is a horizontal compression by 1/|b| units

Question 14

State the reflection transformations

Answer 14

af(x), where a < 0 is a reflection across the x-axis
f(-x) is a reflection across the y-axis

Question 15

State the steps for combining transformations

Answer 15

1. Write the function in factored form y = af(b(x - h)) + k
2. Apply stretches, compressions, reflections first
3. Apply translations/shifts
4. Draw the graph if asked

Question 16

What’s the shortcut for finding a new point after all transformations are applied? (It’s a formula)

Answer 16

To map points:
(x,y) becomes ((1/b) x + h, ay + k)

Question 17

What are invariant points?

Answer 17

Points that stay the same after transformations are applied.

Question 18

What does the transformation notation (p over q) mean?

Answer 18

p is a horizontal shift and q is a vertical shift

Question 19

What does “Mapped to the graph of…” mean?

Answer 19

It means you apply some transformation to one graph to obtain another

Question 20

What is an even function and its properties? State an example.

Answer 20

An even function satisfies f(-x) = f(x) for all x in its domain. It is symmetric about the y-axis, like a mirror reflection. If you plug the negative of x, you get the same output. Polynomial terms are only even degree.

Example: f(x) = x^2

Question 21

What is an odd function and its properties? State an example

Answer 21

An odd function satisfies (f-x) = -f(x) for all x in its domain. It is symmetric about the origin. If you rotate 180 degrees around the origin, it looks the same. If you plug the negative of x, you get the opposite output. If (a, b) is on the graph, (-a,-b) is also on the graph. Polynomial terms are only odd degree.

Example: f(x) = x^3

Question 22

Can a function be both even & odd?

Answer 22

No. The only exception is f(x) = 0

Question 23

What is a composite function?

Answer 23

A new function created by plugging one function into another, where the output of the “inner” function becomes the input for the “outer” function, forming a chain of operations

Notation: f(g(x))

Question 24

What are the domain considerations for composite functions?

Answer 24

1. g(x) must be defined (x is in the domain of g)
2. g(x) must be in the domain of f
3. If some x-values make g(x) invalid for f, restrict the domain

Question 25

What's an inverse function?

Answer 25

An inverse function reverses what the original function does. "Undoing the function". f(x) takes input x, gives output y. The inverse takes input y, gives output x. On a graph, (a, b) on f(x) becomes (b, a) on the inverse

Question 26

What are the properties of an inverse function?

Answer 26

f(inverse) = x inverse(f(x)) = x A function and its inverse cancel out

Question 27

When does an inverse exist?

Answer 27

A function has an inverse if it's one-to-one, meaning each input has a unique output. It must pass the horizontal line test. For example, x^2 has no inverse unless you restrict the domain

Question 28

State the steps to find the inverse

Answer 28

1. Replace f(x) w/ y 2. Swap x and y 3. Solve for y 4. Replace y w/ inverse

Question 29

How does an inverse look graphically?

Answer 29

The inverse is the reflection across the line y = x because you swapped x and y

Question 30

Explain the relationship of domain and range between the original function and the inverse

Answer 30

domain of f = range of inverse range of f = domain of inverse

Question 31

How do you restrict the domain to make a function one-to-one?

Answer 31

Choose the largest possible domain of f that makes the inverse possible

Question 32

State the relationship when two functions are inverses of eachother

Answer 32

If f(x) and g(x) are inverses of eachother: f(g(x)) = x and g(f(x)) = x

Question 33

What's a self-inverse function?

Answer 33

It satisfies f(x) = inverse because it's a function that acts as its own inverse.