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Solutions
58 Videos

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

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

to evaluate the following expressions.

`f(g(0))`

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0.20mins

Q1a

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

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

to evaluate the following expressions.

`g(f(4))`

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Q1b

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

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

to evaluate the following expressions.

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

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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)}`

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0.28mins

Q1d

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

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

to evaluate the following expressions.

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

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Q1e

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

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

to evaluate the following expressions.

`(g\circ g)(2)`

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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)`

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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)`

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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)`

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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)`

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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)`

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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)`

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Q2f

Use the graph of `f`

and `g`

to evaluate each expression.

`f(g(2))`

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Q3a

Use the graph of `f`

and `g`

to evaluate each expression.

`g(f(4))`

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0.19mins

Q3b

Use the graph of `f`

and `g`

to evaluate each expression.

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

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0.17mins

Q3c

Use the graph of `f`

and `g`

to evaluate each expression.

`(f\circ f)(2)`

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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.

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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`

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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`

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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`

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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`

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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`

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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`

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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`

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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`

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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`

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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`

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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`

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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`

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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}`

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0.44mins

Q7a

For each function `h`

, find two functions, `f`

`g`

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

.

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

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0.36mins

Q7b

For each function `h`

, find two functions, `f`

`g`

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

.

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

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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}}`

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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)`

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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}`

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0.39mins

Q7f

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

and `g(x)=x^2`

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

.

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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

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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)`

.

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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)`

.

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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)`

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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.

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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.

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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.

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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.

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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)`

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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)`

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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)`

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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)`

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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)`

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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)`

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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)`

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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)`

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Q14h

`y=f(x)`

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

`g(x)=x+3`

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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)`

?

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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)`

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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)`

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0.41mins

Q14l

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

, and `t=3k-2`

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

.

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0.59mins

Q16a

Express `y`

as a function of `k`

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

, and `t=3k-5`

.

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Q16b

Lectures
8 Videos

Review of Combining functions

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4.03mins

Review of Combining functions

Composition Analogy

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3.36mins

Composition Analogy

Composition with Set of Points

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Composition with Set of Points

Composition with Algegra

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Composition with Algegra

Decomposition of Functions

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Decomposition of Functions

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

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Decomposition example

ex. Find `f, g, h`

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

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Decomposition example 3

Triple Composition

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Triple Composition