The function $f(x) = 2^x + 3$ is defined on $ ext{R}$, the set of real numbers - Scottish Highers Maths - Question 13 - 2015

To sketch $y = f^{-1}(x)$, we reflect the original graph $y = f(x)$ across the line $y = x$. The inverse function graph will pass through the points that correspond to the transformations of points on the original graph.

The coordinates of $P(1, b)$ are $(1, 5)$, and the image of $P$ under the inverse function will be $Q(5, 1)$.

The coordinates of $Q$ can be found by substituting $x = 0$ into the original function:

$f(0) = 2^0 + 3 = 1 + 3 = 4$

Thus, $Q$ corresponds to $(0, 4)$.

To find the image of $R(3, 11)$, we first confirm $f(3)$:

$f(3) = 2^3 + 3 = 8 + 3 = 11$

Now, substituting $(3, 11)$ into the new equation:

$y = 4 - f(x + 1) = 4 - f(4)$

Calculating $f(4)$:

$f(4) = 2^4 + 3 = 16 + 3 = 19$

Thus, $y = 4 - 19 = -15$

The coordinates of the image of $R$ are $(3, -15)$.