Answered: Find all real zeros of the function.…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Finding All Real Zeros of a Function**

**Problem Statement:**
Find all real zeros of the function.

\[ f(x) = -3(x - 4)^2 (x + 7)^3 \]

**Instruction:**
If there is more than one answer, separate them with commas.

**Interactive Input Box:**
There is a text input box labeled "zero(s)" where you can enter your answer.

**Graphical User Interface:**
There is an interface element with options for different layouts or display styles and buttons for confirming or resetting the input.

### Solution Approach:

1. **Understand the Function:**
The function $ f(x) = -3(x - 4)^2 (x + 7)^3 $ is a product of two polynomial factors: $(x - 4)^2$ and $(x + 7)^3$. To find the zeros of the function, we need to determine the values of $x$ that make the function equal to zero.

2. **Set the Function to Zero:**
\[ f(x) = -3(x - 4)^2 (x + 7)^3 = 0 \]
For the product to be zero, at least one of the factors must be zero.

3. **Solve Each Factor:**
- $(x - 4)^2 = 0 $
This factor equals zero when $x = 4$.

- $(x + 7)^3 = 0 $
This factor equals zero when $x = -7$.

4. **Conclusion:**
The real zeros of the function are $x = 4$ and $x = -7$.

**Answer Entry:**
In the input box labeled "zero(s)", you would enter:
\[ 4, -7 \]

This final answer provides the zeros of the function as required.

$For the functions $ f(x) = \frac{4}{x-3} $ and $ g(x) = \frac{1}{x} $, find the composition $ f \circ g $ and simplify your answer as much as possible. Write the domain using interval notation. 1. The screen displays the expression to find the composition: \[ (f \circ g)(x) = f(g(x)) \] 2. Substitution of $ g(x) $ into $ f(x) $: \[ (f \circ g)(x) = f\left(\frac{1}{x}\right) = \frac{4}{\frac{1}{x} - 3} \] 3. To simplify: \[ (f \circ g)(x) = \frac{4}{\frac{1-3x}{x}} = \frac{4x}{1-3x} \] 4. Therefore: \[ (f \circ g)(x) = \frac{4x}{1-3x} \] 5. Next, identify the domain of $ f \circ g $: 6. The domain of $ f(x) $: \[ x - 3 \neq 0 \rightarrow x \neq 3 \] 7. The domain of $ g(x) $: \[ x \neq 0 \] 8. Combining the domains of both functions: \[ g(x) \neq 3 \quad \text{and} \quad x \neq 0 \rightarrow \frac{1}{3} \neq 0 \] Therefore, the domain of $ f \circ g $ in interval notation is: \[ (-\infty, 0) \cup (0, \frac{1}{3}) \cup (\frac{1}{3}, \infty) \] The screen shows a graphical user interface with options to input the answer. There are also radio buttons or checkboxes to select the domain intervals, and a button labeled "Continue" to proceed. **Note:** The data is displayed through an online learning platform or educational website, as evidenced by the McGraw Hill logo and user interface elements such as buttons for submitting assignments and navigation.$

Transcribed Image Text:For the functions $ f(x) = \frac{4}{x-3} $ and $ g(x) = \frac{1}{x} $, find the composition $ f \circ g $ and simplify your answer as much as possible. Write the domain using interval notation. 1. The screen displays the expression to find the composition: \[ (f \circ g)(x) = f(g(x)) \] 2. Substitution of $ g(x) $ into $ f(x) $: \[ (f \circ g)(x) = f\left(\frac{1}{x}\right) = \frac{4}{\frac{1}{x} - 3} \] 3. To simplify: \[ (f \circ g)(x) = \frac{4}{\frac{1-3x}{x}} = \frac{4x}{1-3x} \] 4. Therefore: \[ (f \circ g)(x) = \frac{4x}{1-3x} \] 5. Next, identify the domain of $ f \circ g $: 6. The domain of $ f(x) $: \[ x - 3 \neq 0 \rightarrow x \neq 3 \] 7. The domain of $ g(x) $: \[ x \neq 0 \] 8. Combining the domains of both functions: \[ g(x) \neq 3 \quad \text{and} \quad x \neq 0 \rightarrow \frac{1}{3} \neq 0 \] Therefore, the domain of $ f \circ g $ in interval notation is: \[ (-\infty, 0) \cup (0, \frac{1}{3}) \cup (\frac{1}{3}, \infty) \] The screen shows a graphical user interface with options to input the answer. There are also radio buttons or checkboxes to select the domain intervals, and a button labeled "Continue" to proceed. **Note:** The data is displayed through an online learning platform or educational website, as evidenced by the McGraw Hill logo and user interface elements such as buttons for submitting assignments and navigation.

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