The function f has domain −2 ≤ x < 6 and is linear from (−2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1. (a) Write down the range of f. (b) Find f(0). The function g is defined by g: x → $ \frac{4 + 3x}{5 - x} $, $ x \in \mathbb{R}, \ x \neq 5 $ (c) Find g^{-1}(y). (d) Solve the equation g(f(x)) = 16.

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The function f has domain −2 ≤ x < 6 and is linear from (−2, 10) to (2, 0) and from (2, 0) to (6, 4) - Edexcel - A-Level Maths: Pure - Question 8 - 2013 - Paper 7

Question 8

The function f has domain −2 ≤ x < 6 and is linear from (−2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1. (a) Wr... show full transcript

Worked Solution & Example Answer:The function f has domain −2 ≤ x < 6 and is linear from (−2, 10) to (2, 0) and from (2, 0) to (6, 4) - Edexcel - A-Level Maths: Pure - Question 8 - 2013 - Paper 7

Step 1

Write down the range of f.

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Answer

To determine the range of the function f, we look at the graph provided. The function decreases from (−2, 10) to (2, 0) and then increases again from (2, 0) to (6, 4). The minimum value reached is 0 and the maximum value is 10. Therefore, the range of f is 0 ≤ f(x) ≤ 10.

Step 2

Find f(0).

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Answer

To find f(0), we identify the relevant segment of the function. Since 0 lies within the interval (−2, 2), we use the equation of the line segment from (−2, 10) to (2, 0). The slope (m) is calculated as:

$m = \frac{0 - 10}{2 - (-2)} = -\frac{10}{4} = -\frac{5}{2}$

Using the point-slope form, the equation of the line is given by:

$f(x) = -\frac{5}{2}x + 10$

Substituting x = 0 gives:

$f(0) = -\frac{5}{2}(0) + 10 = 10.$

Step 3

Find g^{-1}(y).

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Answer

To find the inverse of the function g defined by ( g(x) = \frac{4 + 3x}{5 - x} ), we substitute g(x) with y:

$y = \frac{4 + 3x}{5 - x}$

To solve for x, we first cross-multiply:

$y(5 - x) = 4 + 3x$

Expanding yields:

$5y - yx = 4 + 3x$

Rearranging terms leads to:

$yx + 3x = 5y - 4$

Factoring x out results in:

$x(y + 3) = 5y - 4$

Thus, we can express x as:

$x = \frac{5y - 4}{y + 3}$

Hence, the inverse function is:

$g^{-1}(y) = \frac{5y - 4}{y + 3}$

Step 4

Solve the equation g(f(x)) = 16.

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Answer

To solve g(f(x)) = 16, we first substitute f(x) into g:

$g(f(x)) = \frac{4 + 3f(x)}{5 - f(x)} = 16$

Cross-multiplying gives:

$4 + 3f(x) = 16(5 - f(x))$

Simplifying, we have:

$4 + 3f(x) = 80 - 16f(x)$

Rearranging yields:

$3f(x) + 16f(x) = 80 - 4$

Thus,

$19f(x) = 76$

Consequently, we find:

$f(x) = \frac{76}{19} = 4.$

Finally, we must check if f(x) = 4 is within the acceptable domain.

The function f has domain −2 ≤ x < 6 and is linear from (−2, 10) to (2, 0) and from (2, 0) to (6, 4) - Edexcel - A-Level Maths: Pure - Question 8 - 2013 - Paper 7

Worked Solution & Example Answer:The function f has domain −2 ≤ x < 6 and is linear from (−2, 10) to (2, 0) and from (2, 0) to (6, 4) - Edexcel - A-Level Maths: Pure - Question 8 - 2013 - Paper 7

Write down the range of f.

Find f(0).

Find g^{-1}(y).

Solve the equation g(f(x)) = 16.

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