This is a graph of the exponential function $ y = e^x $, where $ e $ is the base of the natural logarithm, approximately equal to 2.71828. The graph is a smooth curve that rapidly increases as the value of $ x $ increases.**Key Features of the Graph:**- **Axes**: The horizontal axis represents the $ x $-values, and the vertical axis represents the $ y $-values.- **Behavior as $ x $ increases**: The graph rises steeply, demonstrating exponential growth.- **Behavior as $ x $ decreases**: As $ x $ becomes more negative, the graph approaches but never quite reaches the $ x $-axis (approaches zero).- **Intercept**: The curve passes through the point (0, 1), indicating that when $ x = 0 $, $ e^x = 1 $.- **Symmetry**: There is no symmetry about the axes, but the function is increasing for all real numbers.This graph is a classic example of an exponential curve, often seen in contexts such as population growth, compound interest, and natural processes. The image contains two sections, each asking for specific X-values based on the positivity or negativity of a function.**Section 1:**- **Instruction:** "Find all the X-values that make the function positive $(f(x) > 0 \text{ when } x = ?)$. Enter your answer in interval notation."- **Guidelines for Input:** Enter $-\infty$ as -INF, $\infty$ as INF, and $\cup$ (union) as U. If none exist, type `none`.- **Expression to Solve:** $f(x) > 0 \text{ when } x \text{ is: }$- **Answer Input Box:** A text box is provided for input.- **Button:** "Submit Answer" with an attempt counter showing "Tries 0/99."**Section 2:**- **Instruction:** "Find all the X-values that make the function negative $(f(x) < 0 \text{ when } x = ?)$. Enter your answer in interval notation."- **Guidelines for Input:** Enter $-\infty$ as -INF, $\infty$ as INF, and $\cup$ (union) as U. If none exist, type `none`.- **Expression to Solve:** $f(x) < 0 \text{ when } x \text{ is: }$- **Answer Input Box:** A text box is provided for input.- **Button:** "Submit Answer" with an attempt counter showing "Tries 0/99."There are no graphs or diagrams involved; the content is focused on determining interval notation solutions for the conditions given.

If f(x)>0 then the curve will be above the x-axis. That occurs for (0,∞) If f(x)<0 then the…

Find all the X-values that make the function positive (f(x) > 0 when x =? ). Enter your answer in interval notation. Enter -0 as -INF, ∞ as INF, and U (union) as U. If none exist, type none. f(x) > 0 when x is : Submit Answer Tries 0/99 Find all the X-values that make the function negative (f(x) < 0 when x =? ). Enter your answer in interval notation. Enter -o as -INF, ∞ as INF, and U (union) as U. If none exist, type none. f(x) < 0 when x is :

Find all the X-values that make the function positive (f(x) > 0 when x =? ). Enter your answer in interval notation. Enter -0 as -INF, ∞ as INF, and U (union) as U. If none exist, type none. f(x) > 0 when x is : Submit Answer Tries 0/99 Find all the X-values that make the function negative (f(x) < 0 when x =? ). Enter your answer in interval notation. Enter -o as -INF, ∞ as INF, and U (union) as U. If none exist, type none. f(x) < 0 when x is :

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

This is a graph of the exponential function $ y = e^x $, where $ e $ is the base of the natural logarithm, approximately equal to 2.71828. The graph is a smooth curve that rapidly increases as the value of $ x $ increases.

**Key Features of the Graph:**

- **Axes**: The horizontal axis represents the $ x $-values, and the vertical axis represents the $ y $-values.
- **Behavior as $ x $ increases**: The graph rises steeply, demonstrating exponential growth.
- **Behavior as $ x $ decreases**: As $ x $ becomes more negative, the graph approaches but never quite reaches the $ x $-axis (approaches zero).
- **Intercept**: The curve passes through the point (0, 1), indicating that when $ x = 0 $, $ e^x = 1 $.
- **Symmetry**: There is no symmetry about the axes, but the function is increasing for all real numbers.

This graph is a classic example of an exponential curve, often seen in contexts such as population growth, compound interest, and natural processes.

The image contains two sections, each asking for specific X-values based on the positivity or negativity of a function.

**Section 1:**
- **Instruction:** "Find all the X-values that make the function positive $(f(x) > 0 \text{ when } x = ?)$. Enter your answer in interval notation."
- **Guidelines for Input:** Enter $-\infty$ as -INF, $\infty$ as INF, and $\cup$ (union) as U. If none exist, type `none`.
- **Expression to Solve:** $f(x) > 0 \text{ when } x \text{ is: }$
- **Answer Input Box:** A text box is provided for input.
- **Button:** "Submit Answer" with an attempt counter showing "Tries 0/99."

**Section 2:**
- **Instruction:** "Find all the X-values that make the function negative $(f(x) < 0 \text{ when } x = ?)$. Enter your answer in interval notation."
- **Guidelines for Input:** Enter $-\infty$ as -INF, $\infty$ as INF, and $\cup$ (union) as U. If none exist, type `none`.
- **Expression to Solve:** $f(x) < 0 \text{ when } x \text{ is: }$
- **Answer Input Box:** A text box is provided for input.
- **Button:** "Submit Answer" with an attempt counter showing "Tries 0/99."

There are no graphs or diagrams involved; the content is focused on determining interval notation solutions for the conditions given.

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