The functions f and g are defined by f: x ↦ 2x + ln 2, x ∈ ℝ, g: x ↦ e^{2x}, x ∈ ℝ - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 9

To sketch the curve of the function $y = gf(x) = 4e^{x}$:

1. Recognize that the function is an exponential function which always cuts the y-axis at (x=0)
2. Substitute (x=0) into the function: $y = 4e^{0} = 4 \Rightarrow (0, 4)$
3. The coordinates where the curve cuts the y-axis are (0, 4).
4. The curve approaches 0 as x approaches negative infinity and increases exponentially as x increases.

The range of the function $gf(x) = 4e^{x}$ is given by the values that the function can take:

• Since the exponential function is always positive, the range is $gf(x) > 0 \Rightarrow (0, +\infty).$

To find the value of x where ( \frac{d}{dx}[gf(x)] = 3 ):

1. First, differentiate the function: $\frac{d}{dx}[gf(x)] = \frac{d}{dx}[4e^{x}] = 4e^{x}.$
2. Set the derivative equal to 3: $4e^{x} = 3$
3. Solve for e^{x}: $e^{x} = \frac{3}{4}$
4. Taking the natural logarithm on both sides: $x = \ln\left(\frac{3}{4}\right)$
5. Calculate (\ln\left(\frac{3}{4}\right)): $x ≈ -0.418$ (to 3 significant figures).