Purchase this Material for $15

You need to sign up or log in to purchase.

Subscribe for All Access
You need to sign up or log in to purchase.

Solutions
58 Videos

Use `f(x)=2x-3`

and `g(x) = 1 -x^2`

to evaluate the following expressions.

`f(g(0))`

Buy to View

0.20mins

Q1a

Use `f(x)=3x-3`

and `g(x) = 1 -x^2`

to evaluate the following expressions.

`g(f(4))`

Buy to View

0.21mins

Q1b

Use `f(x)=3x-3`

and `g(x) = 1 -x^2`

to evaluate the following expressions.

`(f\circ g)(-8)`

Buy to View

0.34mins

Q1c

Use `f(x)=3x-3`

and `g(x) = 1 -x^2`

to evaluate the following expressions.

`\displaystyle{(g\circ g)\left(\frac{1}{2}\right)}`

Buy to View

0.28mins

Q1d

Use `f(x)=3x-3`

and `g(x) = 1 -x^2`

to evaluate the following expressions.

`(f\circ f^{-1})(1)`

Buy to View

0.24mins

Q1e

Use `f(x)=3x-3`

and `g(x) = 1 -x^2`

to evaluate the following expressions.

`(g\circ g)(2)`

Buy to View

0.24mins

Q1f

Given `f=\{(0,1),(1,2),(2,5),(3,10)\}`

and `g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}`

, determine the following values.

`(g\circ f)(2)`

Buy to View

0.24mins

Q2a

Given `f=\{(0,1),(1,2),(2,5),(3,10)\}`

and `g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}`

, determine the following values.

`(f\circ f)(1)`

Buy to View

0.18mins

Q2b

Given `f=\{(0,1),(1,2),(2,5),(3,10)\}`

and `g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}`

, determine the following values.

`(f\circ g)(5)`

Buy to View

0.20mins

Q2c

`f=\{(0,1),(1,2),(2,5),(3,10)\}`

and `g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}`

, determine the following values.

`(f\circ g)(-2)`

Buy to View

0.22mins

Q2d

`f=\{(0,1),(1,2),(2,5),(3,10)\}`

and `g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}`

, determine the following values.

`(f\circ f^{-1})(2)`

Buy to View

0.46mins

Q2e

`f=\{(0,1),(1,2),(2,5),(3,10)\}`

and `g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}`

, determine the following values.

`(g^{-1}\circ f)(1)`

Buy to View

0.43mins

Q2f

Use the graph of `f`

and `g`

to evaluate each expression.

`f(g(2))`

Buy to View

0.23mins

Q3a

Use the graph of `f`

and `g`

to evaluate each expression.

`g(f(4))`

Buy to View

0.19mins

Q3b

Use the graph of `f`

and `g`

to evaluate each expression.

`(g\circ g)(-2)`

Buy to View

0.17mins

Q3c

Use the graph of `f`

and `g`

to evaluate each expression.

`(f\circ f)(2)`

Buy to View

0.23mins

Q3d

For a car travelling at a constant speed of 80 km h, the distance driven, `d`

kilometres, is represented by `d(t)=80t`

, where `t`

is the time in hours. The cost of gasoline, in dollars, for the drive is represented by `C(d)=0.09d`

Determine `C(d(5))`

numerically, and interpret your results.

Buy to View

0.53mins

Q4

In each case, functions `f`

and `g`

are defined for `x\in\mathbb{R}`

. For each pair of functions, determine the expression and the domain of `f(g(x))`

and `g(f(x))`

.

`f(x)=3x^2,g(x)=x-1`

Buy to View

0.55mins

Q5a

In each case, functions `f`

and `g`

are defined for `x\in\mathbb{R}`

. For each pair of functions, determine the expression and the domain of `f(g(x))`

and `g(f(x))`

. Graph each result.

`f(x)=2x^2+x,g(x)=x^2+1`

Buy to View

1.05mins

Q5b

In each case, functions `f`

and `g`

are defined for `x\in\mathbb{R}`

. For each pair of functions, determine the expression and the domain of `f(g(x))`

and `g(f(x))`

. Graph each result.

*`2x^3-3x^2+x-1,g(x)=2x-1`

Buy to View

1.08mins

Q5c

In each case, functions `f`

and `g`

are defined for `x\in\mathbb{R}`

. For each pair of functions, determine the expression and the domain of `f(g(x))`

and `g(f(x))`

. Graph each result.

`f(x)=x^4-x^2,g(x)=x+1`

Buy to View

1.58mins

Q5d

`f`

and `g`

are defined for `x\in\mathbb{R}`

. For each pair of functions, determine the expression and the domain of `f(g(x))`

and `g(f(x))`

. Graph each result.

`f(x)=\sin x,g(x)=4x`

Buy to View

1.25mins

Q5e

`f`

and `g`

are defined for `x\in\mathbb{R}`

. For each pair of functions, determine the expression and the domain of `f(g(x))`

and `g(f(x))`

. Graph each result.

`f(x)=|x|,g(x)=x+5`

Buy to View

1.01mins

Q5f

`f(x)=3x,g(x)=\sqrt{x-4}`

determine the defining equation for

`f\circ g`

and`g\circ f`

determine the domain and range of

`f\circ g`

and`g\circ f`

Buy to View

1.46mins

Q6a

For each of the following,

determine the defining equation for

`f\circ g`

and`g\circ f`

determine the domain and range of

`f\circ g`

and`g\circ f`

`f(x)=\sqrt{x},g(x)=3x+1`

Buy to View

1.26mins

Q6b

For each of the following,

determine the defining equation for

`f\circ g`

and`g\circ f`

determine the domain and range of

`f\circ g`

and`g\circ f`

`f(x)=\sqrt{4-x^2},g(x)=x^2`

Buy to View

3.27mins

Q6c

`f(x)=2^x,g(x)=\sqrt{x-1}`

determine the defining equation for

`f\circ g`

and`g\circ f`

determine the domain and range of

`f\circ g`

and`g\circ f`

Buy to View

1.46mins

Q6d

`f(x)=10^x,g(x)=\log x`

determine the defining equation for

`f\circ g`

and`g\circ f`

determine the domain and range of

`f\circ g`

and`g\circ f`

Buy to View

2.59mins

Q6e

`f(x)=\sin x,g(x)=5^{2x}+1`

determine the defining equation for

`f\circ g`

and`g\circ f`

determine the domain and range of

`f\circ g`

and`g\circ f`

Buy to View

2.34mins

Q6f

For each function `h`

, find two functions, `f`

`g`

, such that `h(x)=f(g(x))`

.

`h(x)=\sqrt{x^2+6}`

Buy to View

0.44mins

Q7a

For each function `h`

, find two functions, `f`

`g`

, such that `h(x)=f(g(x))`

.

`h(x)=(5x-8)^6`

Buy to View

0.36mins

Q7b

For each function `h`

, find two functions, `f`

`g`

, such that `h(x)=f(g(x))`

.

`h(x)=2^{(6x+7)}`

Buy to View

0.40mins

Q7c

For each function `h`

, find two functions, `f`

`g`

, such that `h(x)=f(g(x))`

.

`\displaystyle{h(x)=\frac{1}{x^3-7x+2}}`

Buy to View

0.20mins

Q7d

For each function `h`

, find two functions, `f`

`g`

, such that `h(x)=f(g(x))`

.

`h(x)=\sin^2(10x+5)`

Buy to View

1.09mins

Q7e

For each function `h`

, find two functions, `f`

`g`

, such that `h(x)=f(g(x))`

.

`h(x)=\sqrt[3]{(x+4)^2}`

Buy to View

0.39mins

Q7f

Let `f(x)=2x-1`

and `g(x)=x^2`

. Determine `(f\circ g)(x)`

.

Buy to View

0.19mins

Q8a

Let `f(x)=2x-1`

and `g(x)=x^2`

.

Graph `f,g`

, and `f\circ g`

on the same set of axes

Buy to View

1.05mins

Q8b

Let `f(x)=2x-1`

and `g(x)=x^2`

Describe the graph of `f\circ g`

as a transformation of the graph of `y=g(x)`

.

Buy to View

0.42mins

Q8c

Let `f(x)=2x-1`

and `g(x)=3x+2`

.

- Determine
`f(g(x))`

, and describe its graph as a transformation of`g(x)`

.

Buy to View

1.44mins

Q9a

Let `f(x)=2x-1`

and `g(x)=3x+2`

.

- Determine
`g(f(x))`

, and describe its graph as a transformation of`f(x)`

Buy to View

1.10mins

Q9b

A banquet hall charges $975 to rent a reception room, plus $39.95 per person. Next month, however, the banquet hall will be offering a 20% discount off the total bill. Express this discounted cost as a function of the number of people attending.

Buy to View

0.55mins

Q10

The function `f(x)=0.08x`

represents the sales tax owed on a purchase with a selling price of `x`

dollars, and the function `g(x)=0.75x`

represents the sale price of an item with a price tag of `x`

dollars during a 25% off sale. Write a function that represents the sales tax owed on an item with a price tag of `x`

dollars during a 25% off sale.

Buy to View

1.01mins

Q11

An airplane passes directly over a radar station at time `t=0`

. The plane maintains an altitude of 4 km and is flying at a speed of 560 km/h. Let `d`

represent the distance from the radar station to the plane, and let `s`

represent the horizontal distance travelled by the plane since it passed over the radar station.

**a)** Express `d`

as a function of `s`

, and `s`

as a function of `t`

.

**b)** Use composition to express the distance between the plane and the radar station as a function of time.

Buy to View

1.36mins

Q12

In a vehicle test lab, the speed of a car, `v`

kilometres per hour, at a time of `t`

hours is represented by `v(t)=40+3t+t^2`

. The rate of gasoline consumption of the car, `c`

litres per kilometre, at a speed of `v`

kilometres per hour is represented by `\displaystyle{c(v)=\left(\frac{v}{500}-0.1\right)^2+0.15}`

. Determine algebraically `c(v(t))`

, the rate of gasoline consumption as a function of time. Determine, using technology, the time when the car is running most economically during a 4 h simulation.

Buy to View

1.35mins

Q13

Given the graph of `y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(f\circ g)(x)`

Buy to View

1.05mins

Q14a

Given the graph of `y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(f\circ h)(x)`

Buy to View

0.39mins

Q14b

Given the graph of `y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(f\circ k)(x)`

Buy to View

0.39mins

Q14c

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(f\circ m)(x)`

Buy to View

0.37mins

Q14d

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(f\circ n)(x)`

Buy to View

1.11mins

Q14e

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(f\circ p)(x)`

Buy to View

0.49mins

Q14f

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(g\circ f)(x)`

Buy to View

0.50mins

Q14g

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

*`y=(h\circ f)(x)`

Buy to View

0.36mins

Q14h

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

`g(x)=x+3`

Buy to View

0.55mins

Q14i

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

Which graph is `y=(m\circ f)(x)`

?

Buy to View

0.45mins

Q14j

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(n\circ f)(x)`

Buy to View

0.49mins

Q14k

`y=f(x)`

shown and the functions below, match the correct composition with each graph. Justify your choices.

*i)* `g(x)=x+3`

*ii)* `m(x)=2x`

*iii)* `h(x)=x-3`

*iv)* `n(x)=-0.5x`

*v)* `k(x)=-x`

*vi)* `p(x)=x-4`

`y=(p\circ f)(x)`

Buy to View

0.41mins

Q14l

If `y=3x-2,x=3t+2`

, and `t=3k-2`

, find an expression for `y=f(k)`

.

Buy to View

0.59mins

Q16a

Express `y`

as a function of `k`

if `y=2x+5,x=\sqrt{3t-1}`

, and `t=3k-5`

.

Buy to View

0.42mins

Q16b

Lectures
8 Videos

Review of Combining functions

Buy to View

4.03mins

Review of Combining functions

Composition Analogy

Buy to View

3.36mins

Composition Analogy

Composition with Set of Points

Buy to View

2.18mins

Composition with Set of Points

Composition with Algegra

Buy to View

3.54mins

Composition with Algegra

Decomposition of Functions

Buy to View

1.59mins

Decomposition of Functions

ex. Decompose ```
\displaystyle
y = \frac{1}{\sqrt{1 -x^2}}
```

Buy to View

1.28mins

Decomposition example

ex. Find `f, g, h`

for `f\circ g\circ h(x) = \sqrt{x^2 - 1}`

Buy to View

1.19mins

Decomposition example 3

Triple Composition

Buy to View

2.06mins

Triple Composition