The graph of $f(x) = ext{log}_k x$ is drawn below. B(k; 2) is a point on $f$. 6.1 Calculate the value of $k$. 6.2 Determine the values of $x$ for which $-1 ext{ ≤ } f(x) ext{ ≤ } 2$. 6.3 Write down the equation of $f^{-1}$, the inverse of $f$, in the form $y = ...$ 6.4 For which values of $x$ will $x f'^{-1}(x) < 0$?

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The graph of $f(x) = ext{log}_k x$ is drawn below - NSC Mathematics - Question 6 - 2021 - Paper 1

Question 6

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The graph of $f(x) = ext{log}_k x$ is drawn below. B(k; 2) is a point on $f$. 6.1 Calculate the value of $k$. 6.2 Determine the values of $x$ for which $-1 ext{ ≤... show full transcript

Worked Solution & Example Answer:The graph of $f(x) = ext{log}_k x$ is drawn below - NSC Mathematics - Question 6 - 2021 - Paper 1

Step 1

6.1 Calculate the value of $k$.

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Answer

Given that point B(k; 2) is on the graph of $f(x) = ext{log}_k x$ , we substitute the coordinates into the function:

$2 = ext{log}_k k$

Using the property of logarithms, $ext{log}_k k = 1$ . Therefore, we have:

$2 = 1$

This doesn't hold. Instead, we set up the equation as:

$2 = ext{log}_k k^2$

This implies:

$k^2 = k^2$

Squaring both sides and equating gives:

$4 = k.$ Therefore, $k=16$ .

Step 2

6.2 Determine the values of $x$ for which $-1 ext{ ≤ } f(x) ext{ ≤ } 2$.

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Answer

We start by analyzing the inequalities one at a time.

For the left inequality: $-1 ext{ ≤ } ext{log}_k x$ This is equivalent to: $x ext{ ≥ } k^{-1} = rac{1}{4}.$
For the right inequality: $ext{log}_k x ext{ ≤ } 2$ This can be rewritten as: $x ext{ ≤ } k^2 = 16.$

Combining these gives: $rac{1}{4} ext{ \leq } x ext{ \leq } 16.$

Step 3

6.3 Write down the equation of $f^{-1}$, the inverse of $f$, in the form $y = ...$

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Answer

To find the inverse function, we swap $x$ and $y$ in the equation:

$y = ext{log}_k x.$

Swapping gives:

$x = ext{log}_k y.$

Now, solving for $y$ , we rewrite the equation in exponential form:

$y = k^x.$

Step 4

6.4 For which values of $x$ will $x f'^{-1}(x) < 0$?

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Answer

To determine where the product is negative, we need to examine the behavior of $f'^{-1}(x)$ :

The function $f'(x)$ indicates the rate of change of $f(x)$ : Since $f(x) = ext{log}_k x$ , we have: $f'(x) = rac{1}{x ext{ln} k}.$

The inverse function's derivative: $f'^{-1}(x) = rac{1}{k^x ext{ln} k}.$

Now for the condition: x rac{1}{f'^{-1}(x)} < 0. The product will be negative if:
- Either $x < 0$ (where $f'^{-1}$ is positive)
- Or when both terms are negative or positive, leading to: $x ext{ ∈ } (- ext{∞}; 0).$

The graph of $f(x) = ext{log}_k x$ is drawn below - NSC Mathematics - Question 6 - 2021 - Paper 1

Worked Solution & Example Answer:The graph of $f(x) = ext{log}_k x$ is drawn below - NSC Mathematics - Question 6 - 2021 - Paper 1

6.1 Calculate the value of $k$.

6.2 Determine the values of $x$ for which $-1 ext{ ≤ } f(x) ext{ ≤ } 2$.

6.3 Write down the equation of $f^{-1}$, the inverse of $f$, in the form $y = ...$

6.4 For which values of $x$ will $x f'^{-1}(x) < 0$?

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