Math  /  Algebra

Question1 The functions ff and gg are defined by f(x)=2ln(x+3),x>3f(x)=2 \ln (x+3), x>-3 and g(x)=ekx3,xRg(x)=\mathrm{e}^{k x}-3, x \in \mathbb{R}, where kk is a constant. The function g is the inverse of function f . (a) Determine the value of kk. [4 marks] (b) Sketch the graphs of ff and gg on the same axes. [5 marks]

Studdy Solution

STEP 1

1. The function f(x)=2ln(x+3) f(x) = 2 \ln(x+3) is defined for x>3 x > -3 .
2. The function g(x)=ekx3 g(x) = e^{kx} - 3 is defined for xR x \in \mathbb{R} .
3. The function g g is the inverse of the function f f .
4. We need to find the value of k k .
5. We need to sketch the graphs of f f and g g on the same axes.

STEP 2

1. Find the inverse of f(x) f(x) .
2. Compare the inverse of f(x) f(x) with g(x) g(x) to find k k .
3. Sketch the graph of f(x) f(x) .
4. Sketch the graph of g(x) g(x) .
5. Verify that the graphs are reflections of each other over the line y=x y = x .

STEP 3

To find the inverse of f(x) f(x) , set y=f(x) y = f(x) :
y=2ln(x+3) y = 2 \ln(x+3)
Solve for x x in terms of y y .

STEP 4

Divide both sides by 2:
y2=ln(x+3) \frac{y}{2} = \ln(x+3)

STEP 5

Exponentiate both sides to remove the natural logarithm:
ey2=x+3 e^{\frac{y}{2}} = x + 3

STEP 6

Solve for x x :
x=ey23 x = e^{\frac{y}{2}} - 3
Thus, the inverse function is:
f1(y)=ey23 f^{-1}(y) = e^{\frac{y}{2}} - 3

STEP 7

Since g(x)=ekx3 g(x) = e^{kx} - 3 and g g is the inverse of f f , compare:
ekx3=ex23 e^{kx} - 3 = e^{\frac{x}{2}} - 3
Thus, kx=x2 kx = \frac{x}{2} .

STEP 8

Solve for k k :
k=12 k = \frac{1}{2}

STEP 9

To sketch the graph of f(x)=2ln(x+3) f(x) = 2 \ln(x+3) , note: - The domain is x>3 x > -3 . - The range is all real numbers. - The graph is a logarithmic curve shifted 3 units to the left.

STEP 10

To sketch the graph of g(x)=e12x3 g(x) = e^{\frac{1}{2}x} - 3 , note: - The domain is all real numbers. - The range is y>3 y > -3 . - The graph is an exponential curve shifted 3 units down.

STEP 11

Verify that the graphs of f f and g g are reflections over the line y=x y = x .
- Check that for any point (a,b)(a, b) on f f , the point (b,a)(b, a) is on g g .
The value of k k is:
12 \boxed{\frac{1}{2}}

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