4.1 Calculate the value of a - NSC Mathematics - Question 2 - 2022 - Paper 1

The coordinates of the y-intercept occur where $x = 0$. If the function $g(x)$ is given in a specific form, substitute $x = 0$ into the equation and simplify. The resulting value will be the y-coordinate, and the coordinates will be $(0, g(0))$.

The translation from $g$ to $h$ involves a change in the base of the exponential function. If $g(x)$ has a different base compared to $h(x)$, the translation could imply a vertical stretch or compression depending on the values used. Specifically, changing from base $b$ to ( \frac{1}{3} ) suggests that the graph of $h$ will be narrower and may also reflect changes in its vertical positioning.

To find the inverse of $h(x) = \left( \frac{1}{3} \right)^{x}$, we switch $x$ and $y$:

1. Start with $y = \left( \frac{1}{3} \right)^{x}$

2. Switch to $x = \left( \frac{1}{3} \right)^{y}$

3. Solve for $y$:

   • Taking the logarithm base (\frac{1}{3}) on both sides gives:
   $$y = \log_{\frac{1}{3}}(x)$$
   • Converting the logarithm gives the inverse in an exponential form:
   $$y = \frac{\log(x)}{\log(\frac{1}{3})}.$$