QUESTION 2 The graph of $$ f $$ is given by $$ f(x)=3^{\prime} $$ 2.1 Write down the asymptote of $$ f $$. 2.2 Determine the domain and range of $$ t $$ 2.3 Determine the equation of $$ f^{-1} $$ in the form $$ y=\cdots $$ 2.4 Sketch the graph of $$ f $$ and $$ f^{-1} $$ on the same set of axes. 2.5 (1) Write down the equation of $$ g $$ if $$ g(x)=-f(x)+4 $$ and describe the transformation.

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**2.1 Asymptote of $$ f $$**

Since
$$
f(x)=3^x
$$
is an exponential function, its horizontal asymptote is
$$
y=0.
$$

**2.2 Domain and Range of $$ f $$**

For the exponential function
$$
f(x)=3^x:
$$
- The **domain** is all real numbers, i.e., 
  $$
  \{x \in \mathbb{R}\}.
  $$
- The **range** is all positive real numbers, i.e.,
  $$
  (0, \infty).
  $$

**2.3 Equation of $$ f^{-1} $$**

To find the inverse, start with
$$
y=3^x.
$$
Interchange the roles of $$ x $$ and $$ y $$:
$$
x=3^y.
$$
Take the logarithm with base 3 on both sides:
$$
y=\log_3 x.
$$
Thus, the inverse function is
$$
f^{-1}(x)=\log_3 x,
$$
which can be written in the form
$$
y=\log_3 x.
$$

**2.4 Sketch of $$ f $$ and $$ f^{-1} $$**

- The graph of $$ f(x)=3^x $$ is an increasing exponential curve that:
  - Passes through the point $$(0,1)$$,
  - Approaches $$ y=0 $$ (the horizontal asymptote) as $$ x \to -\infty $$.

- The graph of $$ f^{-1}(x)=\log_3 x $$ is a logarithmic curve that:
  - Passes through the point $$(1,0)$$,
  - Has a vertical asymptote at $$ x=0 $$.

These two graphs are reflections of each other in the line $$ y=x $$.

**2.5 Equation of $$ g $$ and Its Transformation**

Given
$$
g(x)=-f(x)+4,
$$
substitute $$ f(x)=3^x $$ into the equation:
$$
g(x)=-3^x+4.
$$
This transformation can be described as follows:
- **Reflection across the $$ x $$-axis**: The negative sign in front of $$ 3^x $$ reflects the graph of $$ f(x)=3^x $$ over the $$ x $$-axis.
- **Vertical shift upward by 4**: Adding 4 shifts the reflected graph upward by 4 units.

Thus, the final answers are:

- Asymptote of $$ f $$: $$ y=0 $$
- Domain of $$ f $$: $$ \mathbb{R} $$; Range of $$ f $$: $$ (0,\infty) $$
- Inverse function: $$ f^{-1}(x)=\log_3 x $$
- Graphs of $$ f $$ and $$ f^{-1} $$ are reflections in the line $$ y=x $$
- $$ g(x)=-3^x+4 $$ with the transformation being a reflection in the $$ x $$-axis followed by an upward shift of 4 units.
**2.1 Asymptote of $$ f $$:** $$ y = 0 $$ **2.2 Domain and Range of $$ f $$:** - **Domain:** All real numbers ($$ \mathbb{R} $$) - **Range:** Positive real numbers ($$ (0, \infty) $$) **2.3 Equation of $$ f^{-1} $$:** $$ y = \log_3 x $$ **2.4 Sketch of $$ f $$ and $$ f^{-1} $$:** - $$ f(x) = 3^x $$ is an increasing exponential curve passing through $$(0,1)$$ and approaching $$ y = 0 $$ as $$ x \to -\infty $$. - $$ f^{-1}(x) = \log_3 x $$ is a logarithmic curve passing through $$(1,0)$$ with a vertical asymptote at $$ x = 0 $$. - Both graphs are reflections of each other across the line $$ y = x $$. **2.5 Equation of $$ g $$ and Its Transformation:** $$ g(x) = -3^x + 4 $$ *Transformation:* 1. Reflect $$ f(x) = 3^x $$ over the $$ x $$-axis to get $$ -3^x $$. 2. Shift the reflected graph upward by 4 units to obtain $$ g(x) = -3^x + 4 $$.

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