2) T(27; $$ \left.1 \frac{1}{2}\right) $$ is a point on the graph of $$ f(x)=\log a^{x} $$ 2.1) Show that $$ a=9 $$ 2.2) Determine the equation of $$ f^{-1} $$ in the form $$ y=\cdots $$ 3) Determine the equation of $$ g $$ the reflection of $$ f $$ about the $$ x $$ axis 4) Sketch the graphs of $$ f, g $$ and $$ f^{-1} $$ on the same set of axes

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**2.1) Showing that $$a=9$$**

We are given that the point $$T(27;1\frac{1}{2})$$ lies on the graph of  
$$
f(x)=\log_a x.
$$
This means that  
$$
\log_a 27=\frac{3}{2}.
$$
By the definition of logarithms, this equation is equivalent to  
$$
a^{\frac{3}{2}}=27.
$$
To solve for $$a$$, raise both sides to the power $$\frac{2}{3}$$:
$$
\left(a^{\frac{3}{2}}\right)^{\frac{2}{3}}=27^{\frac{2}{3}}.
$$
The left-hand side simplifies since
$$
a^{\frac{3}{2}\cdot\frac{2}{3}}=a^1=a,
$$
and the right-hand side becomes
$$
27^{\frac{2}{3}}=\left(3^3\right)^{\frac{2}{3}}=3^2=9.
$$
Thus, we obtain
$$
a=9.
$$

---

**2.2) Determining the equation of $$f^{-1}$$**

The original function is  
$$
f(x)=\log_{9}x.
$$
The inverse function, $$f^{-1}(x)$$, satisfies
$$
y=f^{-1}(x) \Longleftrightarrow x=f(y).
$$
Starting with
$$
x=\log_{9}y,
$$
we convert from logarithmic to exponential form:
$$
9^x=y.
$$
Thus, the inverse function is given by
$$
f^{-1}(x)=9^x.
$$

---

**3) Equation of $$g$$, the reflection of $$f$$ about the $$x$$-axis**

The reflection of a graph about the $$x$$-axis changes the sign of the $$y$$-coordinate. Given
$$
f(x)=\log_{9}x,
$$
the reflection $$g(x)$$ is
$$
g(x)=-\log_{9}x.
$$

---

**4) Sketching the graphs of $$f$$, $$g$$, and $$f^{-1}$$**

Below is a description of the key features and relative positions of the graphs:

- **Graph of $$f(x)=\log_{9}x$$:**  
  - Domain: $$x>0$$.  
  - $$y$$-intercept: It passes through $$(1,0)$$ because $$\log_{9}1=0$$.  
  - Increasing function.  
  - Passes through $$T(27;\frac{3}{2})$$.

- **Graph of $$f^{-1}(x)=9^x$$:**  
  - Domain: All real numbers $$x$$.  
  - $$y$$-intercept: $$(0,1)$$ because $$9^0=1$$.  
  - Increasing function and exponential growth.  
  - It is the reflection of $$f(x)$$ about the line $$y=x$$.

- **Graph of $$g(x)=-\log_{9}x$$:**  
  - Domain: $$x>0$$.  
  - $$y$$-intercept: $$(1,0)$$ as well (since $$-\log_{9}1=0$$).  
  - Decreasing function (the reflection of $$f(x)$$ about the $$x$$-axis).  
  - The point corresponding to $$T(27;\frac{3}{2})$$ on $$f(x)$$ is reflected to $$(27;-\frac{3}{2})$$.

When sketching these graphs on the same set of axes, note that:

- The graphs of $$f(x)$$ and $$f^{-1}(x)$$ are mirror images of each other about the line $$y=x$$.
- The graph $$g(x)$$ is the vertical reflection of $$f(x)$$.
- All three graphs intersect the point $$(1,0)$$ (for $$f$$ and $$g$$) and $$(0,1)$$ (for $$f^{-1}$$), which are key reference points.

A rough sketch would show:
- The logarithmic curve $$f(x)=\log_{9}x$$ slowly increasing for $$x>1$$ and decreasing for $$0**2.1) Showing that $$a=9$$** Given the point $$T(27;1\frac{1}{2})$$ lies on $$f(x)=\log_a x$$, we have: $$ \log_a 27 = \frac{3}{2} $$ Solving for $$a$$: $$ a^{\frac{3}{2}} = 27 \implies a = 9 $$ --- **2.2) Determining the equation of $$f^{-1}$$** The inverse function is: $$ f^{-1}(x) = 9^x $$ --- **3) Equation of $$g$$, the reflection of $$f$$ about the $$x$$-axis** $$ g(x) = -\log_{9}x $$ --- **4) Sketching the graphs of $$f$$, $$g$$, and $$f^{-1}$$** - **$$f(x) = \log_{9}x$$:** - Passes through $$(1,0)$$ and $$(27, \frac{3}{2})$$. - Increasing function. - **$$f^{-1}(x) = 9^x$$:** - Passes through $$(0,1)$$. - Increasing and exponential growth. - **$$g(x) = -\log_{9}x$$:** - Passes through $$(1,0)$$. - Decreasing function. All graphs intersect at $$(1,0)$$ and $$(0,1)$$.

T(27; is a point on the graph of
2.1) Show that
2.2) Determine the equation of in the form

Determine the equation of the reflection of about the axis

Sketch the graphs of and on the same set of axes

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T(27; is a point on the graph of 2.1) Show that 2.2) Determine the equation of in the form Determine the equation of the reflection of about the axis Sketch the graphs of and on the same set of axes