The diagram below represents the graphs of the functions defined by
$f(x) = \frac{a}{x} + q$
and
g(x) = mx + c - NSC Technical Mathematics - Question 4 - 2022 - Paper 1
Question 4
The diagram below represents the graphs of the functions defined by
$f(x) = \frac{a}{x} + q$
and
g(x) = mx + c.
R(-2; 4) is a point on f.
g is the line o... show full transcript
Worked Solution & Example Answer:The diagram below represents the graphs of the functions defined by
$f(x) = \frac{a}{x} + q$
and
g(x) = mx + c - NSC Technical Mathematics - Question 4 - 2022 - Paper 1
Step 1
4.1.1 Determine the equation of f.
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Answer
To find the equation of f, substitute the point R(-2, 4) into the equation:
4=−2a+q
Rearranging gives us:
4+−2a=q
This means we can say that ( q = 4 + \frac{a}{-2} ).
Next, since the graph of f is symmetric about line g, and g intercepts the y-axis at 2, we find ( q = 2 ) which allows us to solve for a:
2=4+−2a
Thus, ( \frac{a}{-2} = -2 ) resulting in ( a = 4 ).
Therefore, the equation of f is:
f(x)=x4+2
Step 2
4.1.2 Write down the equation of g.
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Answer
Given that g is the line of symmetry, which cuts the y-axis at 2:
g(x)=x+2
Step 3
4.1.3 Write down the range of f.
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Answer
The function f, represented by f(x)=x4+2 has a vertical asymptote at x = 0 and approaches y = 2 but never touches it. Thus, the range of f is:
(−∞,2)∪(2,∞).
Step 4
4.1.4 Determine the coordinates of W.
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Answer
To find the x-intercept W of f, we set f(x) = 0:
0=x4+2
Solving gives:
x4=−2
Thus, ( x = -2 ).
Hence, the coordinates of W are: W(-2, 0).
Step 5
4.1.5 Determine the coordinates of V.
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Answer
Point V is on g, which is determined by setting x equal to -2 (the x-coordinate of point W):
g(−2)=−2+2=0.
