1. Sketch the graph of $$ y=4^{x} $$ and the inverse on the same system of axis Given: $$ f(x)=5^{\mathrm{x}} $$ a) Write down the equation of the inverse of $$ f(x) $$. b) Sketch $$ f $$ and $$ f^{-1} $$ on the same system of axes. c) Write down the equation of the horizontal asymptote of $$ f $$. d) Write down the equation of the vertical asymptote of $$ f-1 $$. e) Give the domain $$ f $$. f) Give the domain of $$ f^{-1} $$.

Question

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Step 1: Sketch the graph of $$ y = 4^x $$ and its inverse

The function $$ y = 4^x $$ is an exponential function that increases rapidly as $$ x $$ increases. Its inverse can be found by switching $$ x $$ and $$ y $$ and solving for $$ y $$.

### Step 2: Find the inverse of $$ f(x) = 5^x $$

To find the inverse, we set $$ y = 5^x $$ and switch $$ x $$ and $$ y $$:

1. Start with $$ x = 5^y $$.
2. Take the logarithm of both sides: $$ \log_5(x) = y $$.

Thus, the inverse function is:

$$
f^{-1}(x) = \log_5(x)
$$

### Step 3: Sketch $$ f $$ and $$ f^{-1} $$

The graph of $$ f(x) = 5^x $$ is an increasing curve that approaches the x-axis (horizontal asymptote) as $$ x $$ approaches negative infinity. The graph of $$ f^{-1}(x) = \log_5(x) $$ is an increasing curve that approaches the y-axis (vertical asymptote) as $$ x $$ approaches zero.

### Step 4: Write down the equations of the asymptotes

**c)** The horizontal asymptote of $$ f(x) = 5^x $$ is:

$$
y = 0
$$

**d)** The vertical asymptote of $$ f^{-1}(x) = \log_5(x) $$ is:

$$
x = 0
$$

### Step 5: Determine the domains

**e)** The domain of $$ f(x) = 5^x $$ is all real numbers:

$$
\text{Domain of } f: (-\infty, \infty)
$$

**f)** The domain of $$ f^{-1}(x) = \log_5(x) $$ is $$ x > 0 $$:

$$
\text{Domain of } f^{-1}: (0, \infty)
$$

### Summary of Results

- **Inverse of $$ f(x) $$**: $$ f^{-1}(x) = \log_5(x) $$
- **Horizontal Asymptote of $$ f $$**: $$ y = 0 $$
- **Vertical Asymptote of $$ f^{-1} $$**: $$ x = 0 $$
- **Domain of $$ f $$**: $$ (-\infty, \infty) $$
- **Domain of $$ f^{-1} $$**: $$ (0, \infty) $$

### Graph Sketch

To sketch the graphs of $$ f(x) = 5^x $$ and $$ f^{-1}(x) = \log_5(x) $$:

1. **Graph of $$ f(x) = 5^x $$**:
   - Passes through (0, 1) since $$ 5^0 = 1 $$.
   - Approaches the x-axis as $$ x $$ goes to negative infinity.

2. **Graph of $$ f^{-1}(x) = \log_5(x) $$**:
   - Passes through (1, 0) since $$ \log_5(1) = 0 $$.
   - Approaches the y-axis as $$ x $$ goes to 0.

Both graphs are reflections of each other across the line $$ y = x $$.

If you need a visual representation, please let me know!
a) The inverse of $$ f(x) = 5^x $$ is $$ f^{-1}(x) = \log_5(x) $$. b) The graphs of $$ f(x) = 5^x $$ and $$ f^{-1}(x) = \log_5(x) $$ are reflections of each other across the line $$ y = x $$. c) The horizontal asymptote of $$ f(x) = 5^x $$ is $$ y = 0 $$. d) The vertical asymptote of $$ f^{-1}(x) = \log_5(x) $$ is $$ x = 0 $$. e) The domain of $$ f(x) = 5^x $$ is all real numbers: $$ (-\infty, \infty) $$. f) The domain of $$ f^{-1}(x) = \log_5(x) $$ is $$ x > 0 $$: $$ (0, \infty) $$.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Calculus Questions