Given: f(x) = \frac{-1}{x-3} + 2 5.1 Write down the equations of the asymptotes of f. 5.2 Write down the domain of f. 5.3 Determine the coordinates of the x-intercept of f. 5.4 Write down the coordinates of the y-intercept of f. 5.5 Draw the graph of f. Clearly show ALL the asymptotes and intercepts with the axes.

NSC Mathematics Functions: Given: f(x) = \frac{-1}{x-3} +

Given: f(x) = \frac{-1}{x-3} + 2 5.1 Write down the equations of the asymptotes of f - NSC Mathematics - Question 5 - 2021 - Paper 1

Question 5

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Given: f(x) = \frac{-1}{x-3} + 2 5.1 Write down the equations of the asymptotes of f. 5.2 Write down the domain of f. 5.3 Determine the coordinates of the x-i... show full transcript

Worked Solution & Example Answer:Given: f(x) = \frac{-1}{x-3} + 2 5.1 Write down the equations of the asymptotes of f - NSC Mathematics - Question 5 - 2021 - Paper 1

Step 1

Write down the equations of the asymptotes of f.

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Answer

To find the asymptotes of the function, we look for vertical and horizontal asymptotes.

The vertical asymptote occurs where the denominator is zero. Thus, set ( x - 3 = 0 ), which gives ( x = 3 ).
The horizontal asymptote can be found by analyzing the end behavior of the function as ( x ) approaches infinity. Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is ( y = 2 ).
Therefore, the equations of the asymptotes are:

( x = 3 ) (vertical asymptote)
( y = 2 ) (horizontal asymptote)

Step 2

Write down the domain of f.

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Answer

The domain of the function excludes values that make the denominator zero. Since ( x = 3 ) results in division by zero, the domain is:
( x \in (-\infty, 3) \cup (3, \infty) )

Step 3

Determine the coordinates of the x-intercept of f.

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Answer

To find the x-intercept, set ( f(x) = 0 ):
[ 0 = \frac{-1}{x-3} + 2 ]
Rearranging gives:
[ \frac{-1}{x-3} = -2 ]
Taking the reciprocal gives:
[ 1 = 2(x - 3) ]
Solving this:
[ 1 = 2x - 6 ]
[ 2x = 7 ]
[ x = \frac{7}{2} ]
Thus, the x-intercept is at ( \left( \frac{7}{2}, 0 \right) ).

Step 4

Write down the coordinates of the y-intercept of f.

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Answer

To find the y-intercept, substitute ( x = 0 ) into the function:
[ f(0) = \frac{-1}{0-3} + 2 = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3} ]
Thus, the y-intercept is at ( (0, \frac{7}{3}) ).

Step 5

Draw the graph of f. Clearly show ALL the asymptotes and intercepts with the axes.

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Answer

The graph of the function ( f(x) ) has the following characteristics:

A vertical asymptote at ( x = 3 )
A horizontal asymptote at ( y = 2 )
The x-intercept at ( \left( \frac{7}{2}, 0 \right) )
The y-intercept at ( (0, \frac{7}{3}) )
To sketch the graph, start with the identified asymptotes, then plot the intercepts. The curve will approach the asymptotes but never touch them, confirming the shape inferred from the rules of rational functions.

Given: f(x) = \frac{-1}{x-3} + 2 5.1 Write down the equations of the asymptotes of f - NSC Mathematics - Question 5 - 2021 - Paper 1

Worked Solution & Example Answer:Given: f(x) = \frac{-1}{x-3} + 2 5.1 Write down the equations of the asymptotes of f - NSC Mathematics - Question 5 - 2021 - Paper 1

Write down the equations of the asymptotes of f.

Write down the domain of f.

Determine the coordinates of the x-intercept of f.

Write down the coordinates of the y-intercept of f.

Draw the graph of f. Clearly show ALL the asymptotes and intercepts with the axes.

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