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The functions f and g are defined by f : x ↦ e^x + 2, x ∈ ℝ g : x ↦ ln x, x > 0 (a) State the range of f - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 5
Question 8
The functions f and g are defined by f : x ↦ e^x + 2, x ∈ ℝ g : x ↦ ln x, x > 0 (a) State the range of f. (b) Find fg(x), giving your answer in its sim... show full transcript
Worked Solution & Example Answer:The functions f and g are defined by f : x ↦ e^x + 2, x ∈ ℝ g : x ↦ ln x, x > 0 (a) State the range of f - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 5
Step 1
State the range of f.
Answer
To determine the range of the function f(x) = e^x + 2, we first note that e^x is defined for all real numbers and its output is always positive. Therefore, the minimum value of e^x is 1 (as x approaches negative infinity). Thus, the minimum value of f(x) is 1 + 2 = 3. Consequently, the range of f is all real numbers greater than 2, which can be expressed as (2, ∞).
Step 2
Step 3
Find the exact value of x for which f(2x + 3) = 6.
Answer
To find x, we start with the equation:
f(2x + 3) = 6.
Replacing f with its definition:
e^{2x + 3} + 2 = 6.
Subtracting 2 from both sides:
e^{2x + 3} = 4.
Taking the natural logarithm of both sides, we have:
2x + 3 = ext{ln }4.
Now, isolating x:
2x = ext{ln }4 - 3
x = rac{ ext{ln }4 - 3}{2}.
Step 4
Find f^{-1}, the inverse function of f, stating its domain.
Answer
To find the inverse function f^{-1}, we first let y = f(x):
y = e^x + 2.
Next, we solve for x:
e^x = y - 2.\
deriving x gives:
x = ext{ln}(y - 2).
Hence, the inverse function is: f^{-1}(y) = ext{ln}(y - 2).
The domain of f^{-1} is the range of f, which is y > 2.
Step 5
On the same axes sketch the curves with equation y = f(x) and y = f^{-1}(x), giving the coordinates of all the points where they cross the axes.
Answer
When sketching the graphs, we highlight the key features of both curves:
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For y = f(x) (which is increasing and starts at (0, 3)):
- It crosses the y-axis at (0, 3).
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For y = f^{-1}(x) (which is also increasing):
- It crosses the x-axis when y = 0, so at the point (2, 0).
Both curves will intersect at the line y = x, indicating symmetry. Ensure the curvature reflects exponential growth for y = f(x) and logarithmic behavior for y = f^{-1}(x).