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Given: $$f(x) = -\frac{1}{4}x^2, \ quad x \leq 0$$ 6.1 Determine the equation of $f^{-1}$ in the form $f^{-1}(x) = .. - NSC Mathematics - Question 6 - 2016 - Paper 1
Question 6
Given: $$f(x) = -\frac{1}{4}x^2, \ quad x \leq 0$$ 6.1 Determine the equation of $f^{-1}$ in the form $f^{-1}(x) = ... 6.2 On the same system of axes, sketch th... show full transcript
Worked Solution & Example Answer:Given: $$f(x) = -\frac{1}{4}x^2, \ quad x \leq 0$$ 6.1 Determine the equation of $f^{-1}$ in the form $f^{-1}(x) = .. - NSC Mathematics - Question 6 - 2016 - Paper 1
Step 1
Step 2
On the same system of axes, sketch the graphs of $f$ and $f^{-1}$. Indicate clearly the intercepts with the axes, as well as another point on the graph of each of $f$ and $f^{-1}$.
Answer
For the graph of :
- The function is a downward-opening parabola with vertex at the origin (0,0).
- The x-intercept is at (0,0).
- Another point is (2,-1) (since at , ).
For the graph of :
- This is also a downward opening curve.
- The y-intercept is at (0,0).
- Another point is (-1, -2), found by substituting into the inverse function.
Step 3
Is $f^{-1}$ a function? Give a reason for your answer.
Answer
Yes, is a function.
No value of in the domain of maps onto more than one -value. This is verified by using the vertical line test, which shows that any vertical line drawn through the graph of will only intersect it at one point.