Given: $h(x) = 2^{-x}$
6.1 Draw a neat sketch of $h$ - NSC Mathematics - Question 6 - 2017 - Paper 1
Question 6
Given: $h(x) = 2^{-x}$
6.1 Draw a neat sketch of $h$.
6.2 Determine the equation of $q$, the graph obtained by reflecting $h$ in the line $y = 0$.
6.3 Write down ... show full transcript
Worked Solution & Example Answer:Given: $h(x) = 2^{-x}$
6.1 Draw a neat sketch of $h$ - NSC Mathematics - Question 6 - 2017 - Paper 1
Step 1
6.1 Draw a neat sketch of $h$.
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Answer
To sketch the function h(x)=2−x, note that it is an exponential decay function. The graph approaches the x-axis as x increases and crosses the y-axis at (0, 1). At x=0, h(0)=20=1.
Step 2
6.2 Determine the equation of $q$, the graph obtained by reflecting $h$ in the line $y = 0$.
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Answer
To reflect h in the line y=0, we take the negative of the function: q(x)=−h(x)=−2−x.
Step 3
6.3 Write down the equation of $h^{-1}$, the inverse of $h$, in the form $y = ...$.
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Answer
To find the inverse, we start with y=2−x. Switching x and y gives us x=2−y. Taking the logarithm base 2 of both sides results in:
y=−extlog2x.
Step 4
6.4 Write down the range of $h$.
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Answer
The range of h(x)=2−x is y>0 or yextextR.
Step 5
6.5 Sketch the graph of $h^{-1}$ on the same set of axes as $h$.
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Answer
The graph of h−1(x)=−extlog2x is a logarithmic function that decreases and passes through the point (1, 0). The asymptote is the x-axis as x approaches 0.
Step 6
6.6 Determine the x-value(s) for which $h^{-1}(x)
eq -3$.
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Answer
To find where h−1(x)=−3, we set up the equation:
−extlog2x=−3
which gives
extlog2x=3.
By exponentiating, we find x=23=8. Therefore, the x-values are:
x=8ext,with0<x<8.
