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Solutions
26 Videos

Match each function with its graph.

```
\displaystyle
\begin{array}{ccccccc}
&a) &h(x)=\displaystyle{\frac{x+4}{2x+5}} &b) & m(x)=\displaystyle{\frac{2x-4}{x-2}} &c) &f(x)=\displaystyle{\frac{3}{x-1}} &d) & g(x)=\displaystyle{\frac{2x-3}{x+2}}
\end{array}
```

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2.57mins

Q1

Consider the function `f(x)=\displaystyle{\frac{3}{x-2}}`

.

**a)** State the equation of the vertical asymptote.

**b)** State the equation of the horizontal asymptote.

**c)** Determine the domain and range.

**d)** Determine the positive and negative intervals.

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4.02mins

Q2

Consider the function `f(x)=\displaystyle{\frac{4x-3}{x+1}}`

.

**a)** State the equation of the vertical asymptote.

**b)** Determine the behaviour(s) of the function
near its vertical asymptote.

**c)** State the equation of the horizontal asymptote.

**d)** Use a table of values to determine the end behaviours of the
function near its horizontal asymptote.

**e)** Determine the domain and range.

**f)** Determine the positive and negative intervals.

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3.54mins

Q3

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

`y=\displaystyle{\frac{2}{x+3}}`

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1.45mins

Q4a

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

`y=\displaystyle{\frac{x-1}{x-5}}`

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3.04mins

Q4b

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

`y=\displaystyle{\frac{2x+1}{2x-1}}`

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1.22mins

Q4c

`y=\displaystyle{\frac{3x+9}{4x+1}}`

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1.12mins

Q4d

For the function describe where the function is increasing or decreasing.

`f(x)=\displaystyle{\frac{3}{x+5}}`

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2.01mins

Q5a

For the function describe where the function is increasing or decreasing.

`f(x)=\displaystyle{\frac{10}{2x-5}}`

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2.40mins

Q5b

For the function describe where the function is increasing or decreasing.

`f(x)=\displaystyle{\frac{x+5}{4x-1}}`

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3.13mins

Q5c

For each function, find

- i. determine the domain,
- ii. intercepts, hole
- iii. asymptotes, and
- iv. positive/negative intervals.
- v. region of increasing or decreasing

`f(x)=\displaystyle{\frac{x+2}{5(x+2)}}`

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0.46mins

Q5d

Read each set of conditions. State which of the equation of a rational function
of the form `f(x)=\displaystyle{\frac{ax+b}{cx+d}}`

meets these conditions

vertical asymptote at `x= -2`

,

horizontal asymptote at `y=0`

;

negative when `x \in (-\infty,-2)`

,

positive when `x \in (-2,\infty)`

;

always decreasing

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0.50mins

Q6a

Read each set of conditions. State which of the equation of a rational function
of the form `f(x)=\displaystyle{\frac{ax+b}{cx+d}}`

meets these conditions.

- vertical asymptote at
`x = -2`

, - horizontal asymptote at
`y=1`

; - x-intercept
`=0`

, y-intercept`=0`

; - positive when
`x \in (-\infty, -2)`

or`(0,\infty)`

, - negative when
`x \in (-2,0)`

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1.19mins

Q6b

Read each set of conditions. State the equation of a rational function
of the form `f(x)=\displaystyle{\frac{ax+b}{cx+d}}`

that meets these conditions.

- hole at
`x=3`

; no vertical asymptote; y-intercept`=0.5`

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0.50mins

Q6c

Read each set of conditions. State the equation of a rational function
of the form `\displaystyle f(x)=\frac{1}{(ax+b)(cx+d)}`

that meets these conditions.
graph.

- vertical asymptotes at
`x=-2`

and`x=6`

, - horizontal asymptote at
`y=0`

; - positive when
`x \epsilon (-\infty,-2)`

or`(6,\infty)`

, negative when`x \epsilon (-2,6)`

; - increasing when
`x \epsilon (-\infty,2)`

, - decreasing when
`x \epsilon (2,\infty)`

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2.58mins

Q6d

Use a graphing calculator to investigate the similarities and
differences in the graphs of rational functions of the form `f(x)=\displaystyle{\frac{8x}{nx+1}}`

, for `n=1,2,4`

, and 8.

Find the equation of the horizontal asymptotes as `n`

approach infinity.

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1.19mins

Q7ab

For `f(x)=\displaystyle{\frac{8x}{nx+1}}`

, if `n`

is negative, how does the function change as the value of `n`

approaches negative infinity? Choose your own values, and use them as examples to support your conclusions.

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0.40mins

Q7c

State the vertical and horizontal asymptote for rational functions `f(x)=\displaystyle{\frac{3x+4}{x-1}}`

and `g(x)=\displaystyle{\frac{x-1}{2x+3}}`

.

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2.01mins

Q8

The function `I(t)=\displaystyle{\frac{15t+25}{t}}`

gives the value of an investment, in thousands of dollars, over `t`

years.

**a)** What is the value of the investment after `2`

years?

**b)** What is the value of the investment after `1`

year?

**c)** What is the value of the investment after `6`

months?

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1.22mins

Q9abc

The function `I(t)=\displaystyle{\frac{15t+25}{t}}`

gives the value of an investment, in thousands of dollars, over `t`

years.

a) Choose a very small value of `t`

(a value near zero). Calculate the
value of the investment at this time. Do you think that the
function is accurate at this time? Why or why not?

b) As time passes, what will the value of the investment approach?

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1.19mins

Q9ef

An amount of chlorine is added to a swimming pool that contains
pure water. The concentration of chlorine, `c`

, in the pool at `t`

hours is
given by `c(t)=\displaystyle{\frac{2t}{2+t}}`

, where `c`

is measured in milligrams per litre. What happens to the concentration of chlorine in the pool during the
24 h period after the chlorine is added?

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1.21mins

Q10

State

- i. the x-intercept
- ii. the equation and vertical asymptotes
- iii. the equation of horizontal asymptotes

for

`f(x)=\displaystyle{\frac{ax+b}{cx+d}}`

in terms of `a, b, c, d`

.

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0.36mins

Q11

Let `f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}`

, `g(x)=\displaystyle{\frac{x^3+8}{x^2+9}},`

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

, and `m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}`

.

Which of these rational functions has a horizontal asymptote?

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0.35mins

Q14a

Let `f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}`

, `g(x)=\displaystyle{\frac{x^3+8}{x^2+9}},`

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

, and `m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}`

.

- Which has an oblique asymptote?

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0.11mins

Q14b

Let `f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}`

, `g(x)=\displaystyle{\frac{x^3+8}{x^2+9}},`

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

, and `m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}`

.

- Which has no vertical asymptote?

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0.40mins

Q14c

Let `f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}`

, `g(x)=\displaystyle{\frac{x^3+8}{x^2+9}}`

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

, and `m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}`

.

Graph `y= m(x)`

, showing the asymptotes and intercepts.

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2.47mins

Q14d