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

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**2.1 Asymptote**  
The function is $$ f(x)=3^{x} $$, an exponential function. It has a horizontal asymptote at  
$$
y=0
$$

**2.2 Domain and Range**  
- **Domain:** Exponential functions are defined for all real numbers, so  
  $$
  \text{Domain}=\{x \in \mathbb{R}\}
  $$  
- **Range:** Since $$ 3^{x}>0 $$ for any $$ x $$, the range is  
  $$
  \text{Range}=(0,\infty)
  $$

**2.3 Equation of $$ f^{-1} $$**  
To find the inverse, start with  
$$
y=3^{x}
$$  
Interchange $$ x $$ and $$ y $$:  
$$
x=3^{y}
$$  
Solve for $$ y $$ by taking the logarithm base 3 of both sides:  
$$
y=\log_{3}x
$$  
Thus, the inverse function is given by  
$$
f^{-1}(x)=\log_{3}x
$$

**2.4 Sketching the Graphs of $$ f $$ and $$ f^{-1} $$**  
- The graph of $$ f(x)=3^{x} $$ is an increasing curve that passes through $$(0,1)$$, with a horizontal asymptote $$ y=0 $$ as $$ x\to -\infty $$.  
- The inverse function $$ f^{-1}(x)=\log_{3}x $$ is the reflection of the graph of $$ f(x) $$ in the line $$ y=x $$. It passes through $$(1,0)$$ and $$(3,1)$$.  
- Both graphs share the symmetry about the line $$ y=x $$.

**2.5 Equation of $$ g $$ and Description of the Transformation**  
Given $$ g(x)=-f(x)+4 $$ and $$ f(x)=3^{x} $$, we have  
$$
g(x)=-3^{x}+4
$$  
The transformations applied to $$ f(x) $$ to obtain $$ g(x) $$ are as follows:  
1. **Reflection in the $$ x $$-axis:** The multiplication by $$-1$$ reflects the graph of $$ f(x) $$ over the $$ x $$-axis, changing $$ 3^{x} $$ into $$-3^{x}$$.  
2. **Vertical Shift Upwards:** Adding $$ 4 $$ shifts the reflected graph upward by $$ 4 $$ units.

Thus, the transformation from $$ f(x)=3^{x} $$ to $$ g(x)=-3^{x}+4 $$ is a reflection in the $$ x $$-axis followed by a vertical shift upward by $$ 4 $$ units.
**2.1 Asymptote** The graph of $$ f(x)=3^{x} $$ has a horizontal asymptote at $$ y=0 $$ **2.2 Domain and Range** - **Domain:** All real numbers $$ \text{Domain}=\{x \in \mathbb{R}\} $$ - **Range:** Positive real numbers $$ \text{Range}=(0,\infty) $$ **2.3 Equation of $$ f^{-1} $$** The inverse function is $$ f^{-1}(x)=\log_{3}x $$ **2.4 Sketching the Graphs** - $$ f(x)=3^{x} $$ is an increasing curve passing through $$(0,1)$$ with a horizontal asymptote at $$ y=0 $$. - $$ f^{-1}(x)=\log_{3}x $$ is the reflection of $$ f(x) $$ over the line $$ y=x $$, passing through $$(1,0)$$ and $$(3,1)$$. **2.5 Equation of $$ g $$ and Transformation** $$ g(x)=-3^{x}+4 $$ Transformation: 1. Reflect $$ f(x) $$ over the $$ x $$-axis to get $$-3^{x}$$. 2. Shift the graph upward by $$ 4 $$ units to obtain $$ g(x)=-3^{x}+4 $$.

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