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 $f$ an... 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 start from the equation of f:
y=bx.
To find the inverse, we interchange x and y:
x=by.
Taking logarithms on both sides gives us:
y = rac{ ext{log}_b(x)}{\log_b} \Rightarrow g(x) = ext{log}_b(x),
which is the required equation in the form of y=....
Step 2
Write down the equation of the line passing through O and R.
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Answer
The points O and R can be determined based on their coordinates. Assuming O is at (0,1) and R is determined from the graph's context, the slope m can be calculated and the line equation would generally be of the form:
y−y1=m(x−x1).
Step 3
Write down the coordinates of point P.
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Answer
The point P is the y-intercept of the curve f(x). Since f(0)=b0=1, the coordinates are P(0, 1).
Step 4
Determine the equation of the line passing through P and T.
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Answer
Using the coordinates of point P(0, 1) and T(1, 0) we can find the slope, m, which equals 1−00−1=−1. The equation of the line can be established as:
y−1=−1(x−0)⇒y=−x+1.
Step 5
Calculate the value of b.
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Answer
At the point R, the value of x can be substituted back into the equation of g:
y=g(21)=logb(21).
From this we can also derive the values through the intersection points yielding b=(21)1/4=41.
