Given:
$g(x) = 2^{x - 1}$ and $h(x) = -\frac{6}{x} - 1$
4.1.1 Write down the equations of the asymptotes of $h$ - NSC Technical Mathematics - Question 4 - 2018 - Paper 1
Question 4
Given:
$g(x) = 2^{x - 1}$ and $h(x) = -\frac{6}{x} - 1$
4.1.1 Write down the equations of the asymptotes of $h$.
4.1.2 Determine the coordinates of the $x$-in... show full transcript
Worked Solution & Example Answer:Given:
$g(x) = 2^{x - 1}$ and $h(x) = -\frac{6}{x} - 1$
4.1.1 Write down the equations of the asymptotes of $h$ - NSC Technical Mathematics - Question 4 - 2018 - Paper 1
Step 1
Write down the equations of the asymptotes of $h$.
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Answer
The function h(x)=−x6−1 has a vertical asymptote at x=0 and a horizontal asymptote at y=−1.
Step 2
Determine the coordinates of the $x$-intercept of $h$.
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Answer
To find the x-intercept, set h(x)=0:
0=−x6−1
Rearranging gives:
x6=−1=>x=−6.
Thus the x-intercept is (−6;0).
Step 3
Sketch the graphs of $g$ and $h$ in the same set of axes.
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Answer
Graphing both functions involves plotting their key features such as intercepts and asymptotes. For g(x), the y-intercept occurs at g(0)=2−1=0.5. Both functions will intersect the axes, and their asymptotes have been drawn clearly. Make sure to label the axes and provide clear marks for significant points.
Step 4
Show that $(-2; 3)$ is a point on the graph of $g$.
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Answer
Plugging x=−2 into g(x):
g(−2)=2−2−1=2−3=81.
Since this does not yield 3, (−2;3) is not a point on the graph of g. Please check calculations or assumptions.
Step 5
Write down the range of $g$.
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Answer
The range of g(x)=2x−1 is (0,∞) since the exponential function can take any positive value but never reaches zero.
Step 6
Write down the domain of $h$.
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Answer
The domain of h(x)=−x6−1 is all real numbers except x=0, written as:
x∈(−∞,0)∪(0,∞).
Step 7
Write down the coordinates of $M$.
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Answer
The coordinates of M are (1;0) as derived from the graph provided.
Step 8
Determine the length of $TR$.
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Answer
The length of TR can be calculated using the distance formula. Measure the vertical distance from point T(1;8) to point R. The coordinates of R will need to be defined based on intersections with the graph of g.
Step 9
Show that $(0; 6)$ are the intercepts of $f$.
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Answer
To find the intercepts of f(x)=a(x+p)2+q, set both x and y values to be zero and solve accordingly. Given that (0;6) is supposed to be an intercept, substituting into the equation will confirm the relationship.
Step 10
Show that the graph of $f$ is defined by $f(x) = -2(x - 1)(x - 3)$.
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Answer
Expanding f(x)=−2(x−1)(x−3):
f(x)=−2(x2−4x+3)=−2x2+8x−6
confirming the values obtained through intercept calculations.
Step 11
Find $K(-1; 0)$.
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Answer
Given that K is an intercept, substituting x=−1 into the function will yield the corresponding y, confirming the coordinates at intersection points.
Step 12
Determine $x = (-1; 0)$.
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Answer
x can be noted as (−1;0) and utilized in further calculations concerning intersection or verification in algebraic terms.
