In the sketch below, P is the y-intercept of the graph of $f(x) = b^x$ - NSC Mathematics - Question 6 - 2017 - Paper 1
Question 6
In the sketch below, P is the y-intercept of the graph of $f(x) = b^x$. T is the x-intercept of graph $g$, the inverse of $f$. R is the point of intersection of $f$ ... show full transcript
Worked Solution & Example Answer:In the sketch below, P is the y-intercept of the graph of $f(x) = b^x$ - NSC Mathematics - Question 6 - 2017 - Paper 1
Step 1
Determine the equation of g (in terms of b) in the form y = ...
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Answer
To find the equation of the inverse function g, we interchange x and y in the equation of f:
y=bx
Becomes:
x=by
Taking the logarithm base b of both sides gives:
y = rac{ ext{log}_b(x)}{1}
Thus, the equation of g is:
g:y=extlogb(x).
Step 2
Write down the equation of the line passing through O and R.
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Answer
The coordinates of O are (0, 1) and the coordinates of R can be derived from the intersection of f and g. Assuming they intersect at (1, 1/2), the slope (m) of the line OR can be calculated as:
m = rac{1/2 - 1}{1 - 0} = - rac{1}{2}
Using the point-slope form of the line equation through point O:
y - 1 = - rac{1}{2}(x - 0)
This simplifies to:
y = - rac{1}{2}x + 1.
Step 3
Write down the coordinates of point P.
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Answer
The point P is the y-intercept of the graph of f(x)=bx. Since the curve intersects the y-axis when x=0, we can substitute:
When x=0oy=b0=1.
Thus, the coordinates of point P are (0, 1).
Step 4
Determine the equation of the line passing through P and T.
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Answer
We need the coordinates of T, which is (1, 0) as the x-intercept of g. Using points P (0, 1) and T (1, 0), we can find the slope:
m = rac{0 - 1}{1 - 0} = -1
Using the point-slope form from point P:
y−1=−1(x−0)
This gives us:
y=−x+1.
Step 5
Calculate the value of b.
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Answer
To find the value of b, we use the coordinates of R, where y = rac{1}{2}. Substituting into the equation:
