For the constant $k$, where $k > 1$, the functions $f$ and $g$ are defined by $f: x o ext{ln}(x + k), ext{ for } x > -k,$ g: x o |2x - k|, ext{ for } x ext{ in } ext{R}. (a) On separate axes, sketch the graph of $f$ and the graph of $g$. On each sketch state, in terms of $k$, the coordinates of points where the graph meets the coordinate axes. (b) Write down the range of $f$. (c) Find $f(g(k/4))$ in terms of $k$, giving your answer in its simplest form. (d) The curve $C$ has equation $y = f(x)$. The tangent to $C$ at the point with $x$-coordinate 3 is parallel to the line with equation $9y = 2x + 1$. Find the value of $k$.

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For the constant $k$, where $k > 1$, the functions $f$ and $g$ are defined by $f: x o ext{ln}(x + k), ext{ for } x > -k,$ g: x o |2x - k|, ext{ for } x ext{ in } ext{R} - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Question 8

For-the-constant-$k$,-where-$k->-1$,-the-functions-$f$-and-$g$-are-defined-by--$f:-x--o--ext{ln}(x-+-k),--ext{-for-}-x->--k,$--g:-x--o-|2x---k|,--ext{-for-}-x--ext{-in-}--ext{R}-Edexcel-A-Level Maths Pure-Question 8-2006-Paper 4.png

For the constant $k$, where $k > 1$, the functions $f$ and $g$ are defined by $f: x o ext{ln}(x + k), ext{ for } x > -k,$ g: x o |2x - k|, ext{ for } x ext{ ... show full transcript

Worked Solution & Example Answer:For the constant $k$, where $k > 1$, the functions $f$ and $g$ are defined by $f: x o ext{ln}(x + k), ext{ for } x > -k,$ g: x o |2x - k|, ext{ for } x ext{ in } ext{R} - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Step 1

Sketch the graph of $f$ and the graph of $g$

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Answer

To sketch the graphs of the functions:

Graph of $f$ : The function $f(x) = ext{ln}(x + k)$ will have a vertical asymptote at $x = -k$ . Its graph intersects the y-axis at $(0, ext{ln}(k))$ . The graph will increase as $x$ increases, approaching infinity as $x$ approaches infinity.
Graph of $g$ : The function $g(x) = |2x - k|$ is a V-shaped graph. It intersects the x-axis at points where $2x - k = 0$ (i.e., $x = rac{k}{2}$ ). The vertex of the graph occurs at this point, with a slope of 2 as $x > rac{k}{2}$ and -2 as $x < rac{k}{2}$ . The y-intercept can be found by evaluating $g(0) = |2(0) - k| = | - k | = k$ .

Thus, the graphs are as follows:

$f$ : intersects y-axis at $(0, ext{ln}(k))$ and has a vertical asymptote at $x = -k$ .
$g$ : intersects x-axis at $ig( rac{k}{2}, 0ig)$ and y-axis at $(0, k)$ .

Step 2

Write down the range of $f$.

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Answer

The range of $f(x) = ext{ln}(x + k)$ , given that $k > 1$ , is all real numbers, i.e.,

$(- ext{infinity}, + ext{infinity}).$

Step 3

Find $f(g(k/4))$ in terms of $k$, giving your answer in its simplest form.

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Answer

First, compute $g(k/4)$ :

$g(k/4) = |2(k/4) - k| = |k/2 - k| = |- rac{k}{2}| = rac{k}{2}.$

Now, substitute into $f$ :

$f(g(k/4)) = figg( rac{k}{2}igg) = ext{ln}igg( rac{k}{2} + kigg) = ext{ln}igg( rac{3k}{2}igg).$

Step 4

Find the value of $k$.

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Answer

To find $k$ , first determine the slope of the line given:

The line $9y = 2x + 1$ can be rewritten in slope-intercept form:

$y = rac{2}{9}x + rac{1}{9}.$

Thus, the slope is $rac{2}{9}$ . Now, we need the derivative of $f(x)$ to obtain the slope of the tangent at $x = 3$ :

$f'(x) = rac{1}{x + k}.$

At $x = 3$ :

$f'(3) = rac{1}{3 + k}.$

Setting it equal to the slope of the line:

$rac{1}{3 + k} = rac{2}{9}.$

Cross-multiplying yields:

9 = 6 + 2k \ 3 = 2k \ k = rac{3}{2} = 1.5.$$

For the constant $k$, where $k > 1$, the functions $f$ and $g$ are defined by $f: x o ext{ln}(x + k), ext{ for } x > -k,$ g: x o |2x - k|, ext{ for } x ext{ in } ext{R} - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Worked Solution & Example Answer:For the constant $k$, where $k > 1$, the functions $f$ and $g$ are defined by $f: x o ext{ln}(x + k), ext{ for } x > -k,$ g: x o |2x - k|, ext{ for } x ext{ in } ext{R} - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Sketch the graph of $f$ and the graph of $g$

Write down the range of $f$.

Find $f(g(k/4))$ in terms of $k$, giving your answer in its simplest form.

Find the value of $k$.

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