Writing and Graphing the Reflection of an Exponential Function
Find and graph the equation for a function, , that reflects about the -axis. State
its domain, range, and asymptote.

Ask by Marsh Morrison. in the United States

Mar 07,2025

Question

Writing and Graphing the Reflection of an Exponential Function
Find and graph the equation for a function, , that reflects about the -axis. State
its domain, range, and asymptote.

Ask by Marsh Morrison. in the United States

Mar 07,2025

UpStudy AI · Accepted Answer

Below is a step‐by‐step solution in markdown with LaTeX formulas:

---

### Step 1. Write the Equation for $$ g(x) $$

A reflection of the function $$ f(x) $$ about the $$ x $$-axis is found by multiplying the function by $$-1$$. Given

$$
f(x) = \left(\frac{1}{4}\right)^x,
$$

the reflected function is

$$
g(x) = -\left(\frac{1}{4}\right)^x.
$$

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### Step 2. Determine the Domain

For the original function $$ \left(\frac{1}{4}\right)^x $$, the domain is all real numbers. Since reflecting about the $$ x $$-axis does not affect the domain, we have:

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

---

### Step 3. Determine the Range

The function $$ \left(\frac{1}{4}\right)^x $$ is always positive for any real $$ x $$ (i.e., its range is $$ (0, \infty) $$). Reflecting it about the $$ x $$-axis (by multiplying by $$-1$$) makes all values negative. Therefore, the range of $$ g(x) $$ is:

$$
\text{Range: } (-\infty, 0).
$$

---

### Step 4. Identify the Asymptote

The original function $$ \left(\frac{1}{4}\right)^x $$ has a horizontal asymptote at $$ y=0 $$ because as $$ x \to \infty $$, the function tends to $$ 0 $$. Reflecting it over the $$ x $$-axis keeps the horizontal asymptote unchanged:

$$
\text{Asymptote: } y=0.
$$

---

### Step 5. Graphing the Function

On the graph:

- **$$ g(x) = -\left(\frac{1}{4}\right)^x $$** is an exponential function.
- For $$ x=0 $$, $$ g(0) = -\left(\frac{1}{4}\right)^0 = -1 $$.
- As $$ x \to \infty $$, $$ \left(\frac{1}{4}\right)^x \to 0 $$ so $$ g(x) \to 0 $$ from below (but never reaches $$ 0 $$).
- As $$ x \to -\infty $$, $$ \left(\frac{1}{4}\right)^x \to \infty $$ so $$ g(x) \to -\infty $$.

Thus, the graph starts very low for very negative $$ x $$, crosses the point $$ (0, -1) $$, and then approaches the horizontal line $$ y=0 $$ without ever touching it, remaining below it.

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### Final Answer

- **Function:** $$\displaystyle g(x) = -\left(\frac{1}{4}\right)^x$$
- **Domain:** $$\displaystyle (-\infty, \infty)$$
- **Range:** $$\displaystyle (-\infty, 0)$$
- **Horizontal Asymptote:** $$ y = 0 $$

This completes the step-by-step process for writing and graphing the reflection of the given exponential function.
$$ g(x) = -\left(\frac{1}{4}\right)^x $$ - **Domain:** All real numbers - **Range:** All negative real numbers - **Asymptote:** $$ y = 0 $$ The graph of $$ g(x) $$ is an exponential function that starts very low for negative $$ x $$, crosses the point $$ (0, -1) $$, and approaches the horizontal line $$ y = 0 $$ without ever touching it, remaining below it.

Writing and Graphing the Reflection of an Exponential Function
Find and graph the equation for a function, , that reflects about the -axis. State
its domain, range, and asymptote.

Upstudy AI Solution

Answer

Solution

Mind Expander

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Writing and Graphing the Reflection of an Exponential Function Find and graph the equation for a function, , that reflects about the -axis. State its domain, range, and asymptote.

Upstudy AI Solution

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its domain, range, and asymptote.