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Solutions
18 Videos

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

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

. Determine a simplified algebraic model for the composite function.

`y = f(g(x))`

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0.36mins

Q1a

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

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

. Determine a simplified algebraic model for the composite function.

`y = g(f(x))`

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0.31mins

Q1b

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

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

. Determine a simplified algebraic model for the composite function.

`y = f(f(x))`

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0.30mins

Q1c

`f(x) = -x + 2`

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

. Determine a simplified algebraic model for the composite function.

`y = g(g(x))`

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0.24mins

Q1d

`f(x) = -x + 2`

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

. Determine a simplified algebraic model for the composite function.

`y = f^{-1}(f(x))`

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0.32mins

Q1e

Graph each functions given `f(x) = -x + 2`

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

.

**a)** `y = f(g(x))`

**b)** `y = g(f(x))`

**c)** `y = f(f(x))`

**d)** `y = g(g(x))`

**e)** `y = f^{-1}(f(x))`

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2.23mins

Q2

Let `f(x) = x^2 + 2x -4`

and `g(x) = \frac{1}{x + 1}`

.

Evaluate `g(f(0))`

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0.26mins

Q4a

Let `f(x) = x^2 + 2x -4`

and `g(x) = \frac{1}{x + 1}`

.

Evaluate `f(g(-2))`

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0.21mins

Q4b

Let `f(x) = x^2 + 2x -4`

and `g(x) = \frac{1}{x + 1}`

.

Show that `g(f(x))`

is undefined for `x = 1`

and `x = -3`

.

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0.35mins

Q4c

Is `f(g(x)) = g(f(x))`

true for all functions `f(x)`

and `g(x)`

? Show your work.

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1.01mins

Q8

Let `f(x) = x^3`

.

**a)** Determine `f^{-1}(x)`

.

**b)** Determine `f(f^{-1}(x))`

.

**c)** Determine `f^{-1}(f(x))`

.

**d)** Compare your answer to parts b) and c). Describe what you notice.

e) Determine `f(f^{-1}(3)), f(f^{-1}(5))`

, and `f(f^{-1}(-1))`

. What do you notice.

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1.28mins

Q9

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

, or is `(f \circ g)^{-1}(x) = (g^{-1}\circ f^{-1})(x)`

, or is neither true? Verify using examples.

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2.32mins

Q14

Let `f(x) = x^2, g(x) = x - 2`

, and `h(x) = \frac{1}{x}`

.

**a)** Determine a simplified algebraic model for each composite function.

- i.
`f(g(x))`

- ii.
`h(g(x))`

- iii.
`g^{-1}(h(x))`

**b)** Evaluate `f(h(2))`

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1.02mins

Q17

Let `f(x) = x^2 - 9`

and `g(x) = \frac{1}{x}`

.

- Determine the domain and range of
`y = f(g(x))`

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2.00mins

Q19a

Let `f(x) = x^2 - 9`

and `g(x) = \frac{1}{x}`

.

- Determine the domain and range of
`y = g(f(x))`

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1.43mins

Q19b

Given that `f(x) =\dfrac{ax + b}{cx + d}`

, find `f^{-1}(x)`

.

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1.12mins

Q21

Find all real numbers `x`

such that `\sqrt{1 - \sqrt{1 - x}} = x`

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0.00mins

Q23

The sum of two numbers is 7 and their product is 25 . Determine the sum of their reciprocals.

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0.30mins

Q24

Lectures
15 Videos

Identifying parts of Composite Functions

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1.32mins

Identifying parts of Composite Functions

Composition using set of points

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2.49mins

Composition using set of points

Domain of Composite Function

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3.26mins

Domain of Composite Function

Domain of Composite Function

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4.46mins

Domain of Composite Function

Domain of Composite Function, challenging

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2.04mins

Domain of Composite Function, challenging

Domain of Composite Function ex2

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1.33mins

Domain of Composite Function ex2

Below is an example of Even Function.

Algebraically for `y = f(x)`

needs to satisfy

```
\displaystyle
f(-x) = -f(x)
```

to be an even function.

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4.49mins

Even Function

Odd Function

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3.35mins

Odd Function

Composition with algegra

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3.54mins

Composition with algegra

Decomposition

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1.59mins

Decomposition

Triple composition

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2.06mins

Triple composition

Decomposition example

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1.28mins

Decomposition example

Decomposition example 3

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1.19mins

Decomposition example 3

Sketching degree 1 Rational Function

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2.03mins

Sketching degree 1 Rational Function