The graph of $g(x)=a \left( \frac{1}{3} \right) ^x + 7$ passes through point E(-2; 10) - NSC Mathematics - Question 4 - 2022 - Paper 1

To find the y-intercept, set $x = 0$ in the function:

$y = g(0) = a \left( \frac{1}{3} \right)^0 + 7$

Substituting $a = \frac{1}{3}$:

$y = \frac{1}{3} \cdot 1 + 7 = \frac{1}{3} + 7 = \frac{22}{3}$

Thus, the coordinates of the y-intercept are: $(0, \frac{22}{3})$.

The translation from $g$ to $h$ involves moving the graph 1 unit to the right and 7 units downwards.

The function $h(x)$ is defined as $h(x) = \left( \frac{1}{3} \right)^x$. To find the inverse:

1. Swap $x$ and $y$: $x = \left( \frac{1}{3} \right)^y$

2. Solve for $y$ by taking logarithms: $y = -\log_{\frac{1}{3}}(x)$

Thus, the equation of the inverse is: $$h^{-1}(x) = -\log_{\frac{1}{3}}(x)$.