Consider The Function F ( X ) = 4 − X − 2 F(x) = 4^{-x} - 2 F ( X ) = 4 − X − 2 .4.1 Calculate The Coordinates Of The Intercepts Of F F F With The Axes.4.2 Write Down The Equation Of The Asymptote Of F F F .4.3 Sketch The Graph Of F F F .4.4 Write Down The

4.1 Calculating the Coordinates of the Intercepts of $f$ with the Axes

To find the coordinates of the intercepts of the function $f(x) = 4^{-x} - 2$ with the axes, we need to consider two cases: when the function intersects the x-axis and when it intersects the y-axis.

Intercept with the x-axis:

When the function intersects the x-axis, the value of $y$ is equal to zero. Therefore, we set $f(x) = 0$ and solve for $x$.

$$4^{-x} - 2 = 0$$

Adding 2 to both sides of the equation, we get:

$$4^{-x} = 2$$

Taking the logarithm of both sides with base 4, we get:

$$-x = \log_4(2)$$

Multiplying both sides by -1, we get:

$$x = -\log_4(2)$$

Using the change of base formula, we can rewrite the equation as:

$$x = \frac{\log(2)}{\log(4)}$$

Since $\log(4) = 2\log(2)$, we can simplify the equation to:

$$x = \frac{\log(2)}{2\log(2)}$$

$$x = \frac{1}{2}$$

Therefore, the function intersects the x-axis at the point $(\frac{1}{2}, 0)$.

Intercept with the y-axis:

When the function intersects the y-axis, the value of $x$ is equal to zero. Therefore, we substitute $x = 0$ into the function and solve for $y$.

$$f(0) = 4^{-0} - 2$$

$$f(0) = 1 - 2$$

$$f(0) = -1$$

Therefore, the function intersects the y-axis at the point $(0, -1)$.

4.2 Writing Down the Equation of the Asymptote of $f$

The equation of the asymptote of a function can be found by considering the behavior of the function as $x$ approaches infinity or negative infinity.

In this case, the function $f(x) = 4^{-x} - 2$ approaches $-2$ as $x$ approaches infinity or negative infinity. Therefore, the equation of the asymptote is:

$$y = -2$$

4.3 Sketching the Graph of $f$

To sketch the graph of the function $f(x) = 4^{-x} - 2$, we need to consider the following:

• The function intersects the x-axis at the point $(\frac{1}{2}, 0)$.
• The function intersects the y-axis at the point $(0, -1)$.
• The equation of the asymptote is $y = -2$.

Using this information, we can sketch the graph of the function as follows:

The graph of the function $f(x) = 4^{-x} - 2$ is a curve that approaches the asymptote $y = -2$ as $x$ approaches infinity or negative infinity. The curve intersects the x-axis at the point $(\frac{1}{2}, 0)$ and the y-axis at the point $(0, -1)$.

4.4 Writing Down the Domain and Range of $f$

The domain of a function is the set of all possible input values for which the function is defined. In this case, the function $f(x) = 4^{-x} - 2$ is defined for all real numbers $x$. Therefore, the domain of the function is:

$$(-\infty, \infty)$$

The range of a function is the set of all possible output values for which the function is defined. In this case, the function $f(x) = 4^{-x} - 2$ approaches $-2$ as $x$ approaches infinity or negative infinity. Therefore, the range of the function is:

$$(-\infty, -2)$$

4.5 Writing Down the Derivative of $f$

The derivative of a function is a measure of how fast the function changes as the input changes. In this case, we can find the derivative of the function $f(x) = 4^{-x} - 2$ using the chain rule.

$$f'(x) = -\ln(4) \cdot 4^{-x}$$

4.6 Writing Down the Second Derivative of $f$

The second derivative of a function is a measure of how fast the first derivative changes as the input changes. In this case, we can find the second derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the first derivative.

$$f''(x) = \ln(4)^2 \cdot 4^{-x}$$

4.7 Writing Down the Third Derivative of $f$

The third derivative of a function is a measure of how fast the second derivative changes as the input changes. In this case, we can find the third derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the second derivative.

$$f'''(x) = -\ln(4)^3 \cdot 4^{-x}$$

4.8 Writing Down the Fourth Derivative of $f$

The fourth derivative of a function is a measure of how fast the third derivative changes as the input changes. In this case, we can find the fourth derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the third derivative.

$$f''''(x) = \ln(4)^4 \cdot 4^{-x}$$

4.9 Writing Down the Fifth Derivative of $f$

The fifth derivative of a function is a measure of how fast the fourth derivative changes as the input changes. In this case, we can find the fifth derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the fourth derivative.

$$f'''''(x) = -\ln(4)^5 \cdot 4^{-x}$$

4.10 Writing Down the Sixth Derivative of $f$

The sixth derivative of a function is a measure of how fast the fifth derivative changes as the input changes. In this case, we can find the sixth derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the fifth derivative.

$$f''''''(x) = \ln(4)^6 \cdot 4^{-x}$$

4.11 Writing Down the Seventh Derivative of $f$

The seventh derivative of a function is a measure of how fast the sixth derivative changes as the input changes. In this case, we can find the seventh derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the sixth derivative.

$$f''''''](x) = -\ln(4)^7 \cdot 4^{-x}$$

4.12 Writing Down the Eighth Derivative of $f$

The eighth derivative of a function is a measure of how fast the seventh derivative changes as the input changes. In this case, we can find the eighth derivative of the function $f(x) = 4^{-x} - 2$ by differentiating the seventh derivative.

f'''''''](x) = \ln(4)^8 \cdot 4^{-x}$<br/> **Q&A: Understanding the Function and Its Graphical Representation** ===========================================================

Q1: What is the function $f(x) = 4^{-x} - 2$?

A1: The function $f(x) = 4^{-x} - 2$ is an exponential function that involves the base 4 and a constant term of -2.

Q2: What are the coordinates of the intercepts of $f$ with the axes?

A2: The function intersects the x-axis at the point $(\frac{1}{2}, 0)$ and the y-axis at the point $(0, -1)$.

Q3: What is the equation of the asymptote of $f$?

A3: The equation of the asymptote of $f$ is $y = -2$.

Q4: How do you sketch the graph of $f$?

A4: To sketch the graph of $f$, you need to consider the following:

• The function intersects the x-axis at the point $(\frac{1}{2}, 0)$.
• The function intersects the y-axis at the point $(0, -1)$.
• The equation of the asymptote is $y = -2$.

Q5: What is the domain of $f$?

A5: The domain of $f$ is $(-\infty, \infty)$.

Q6: What is the range of $f$?

A6: The range of $f$ is $(-\infty, -2)$.

Q7: How do you find the derivative of $f$?

A7: You can find the derivative of $f$ using the chain rule.

Q8: What is the derivative of $f$?

A8: The derivative of $f$ is f&#x27;(x) = -\ln(4) \cdot 4^{-x}.

Q9: How do you find the second derivative of $f$?

A9: You can find the second derivative of $f$ by differentiating the first derivative.

Q10: What is the second derivative of $f$?

A10: The second derivative of $f$ is f&#x27;&#x27;(x) = \ln(4)^2 \cdot 4^{-x}.

Q11: How do you find the third derivative of $f$?

A11: You can find the third derivative of $f$ by differentiating the second derivative.

Q12: What is the third derivative of $f$?

A12: The third derivative of $f$ is f&#x27;&#x27;&#x27;(x) = -\ln(4)^3 \cdot 4^{-x}.

Q13: How do you find the fourth derivative of $f$?

A13: You can find the fourth derivative of $f$ by differentiating the third derivative.

Q14: What is the fourth derivative of $f$?

A14: The fourth derivative of $f$ is f&#x27;&#x27;&#x27;&#x27;(x) = \ln(4)^4 \cdot 4^{-x}.

Q15: How do you find the fifth derivative of $f$?

A15: You can find the fifth derivative of $f$ by differentiating the fourth derivative.

Q16: What is the fifth derivative of $f$?

A16: The fifth derivative of $f$ is f&#x27;&#x27;&#x27;&#x27;&#x27;(x) = -\ln(4)^5 \cdot 4^{-x}.

Q17: How do you find the sixth derivative of $f$?

A17: You can find the sixth derivative of $f$ by differentiating the fifth derivative.

Q18: What is the sixth derivative of $f$?

A18: The sixth derivative of $f$ is f&#x27;&#x27;&#x27;&#x27;&#x27;&#x27;(x) = \ln(4)^6 \cdot 4^{-x}.

Q19: How do you find the seventh derivative of $f$?

A19: You can find the seventh derivative of $f$ by differentiating the sixth derivative.

Q20: What is the seventh derivative of $f$?

A20: The seventh derivative of $f$ is f&#x27;&#x27;&#x27;&#x27;&#x27;&#x27;](x) = -\ln(4)^7 \cdot 4^{-x}.

Q21: How do you find the eighth derivative of $f$?

A21: You can find the eighth derivative of $f$ by differentiating the seventh derivative.

Q22: What is the eighth derivative of $f$?

A22: The eighth derivative of $f$ is f&#x27;&#x27;&#x27;&#x27;&#x27;&#x27;&#x27;](x) = \ln(4)^8 \cdot 4^{-x}.

Q23: What is the significance of the derivatives of $f$?

A23: The derivatives of $f$ provide information about the behavior of the function, including its rate of change and concavity.

Q24: How do you use the derivatives of $f$ in real-world applications?

A24: The derivatives of $f$ can be used in a variety of real-world applications, including modeling population growth, understanding chemical reactions, and analyzing economic systems.

Q25: What are some common mistakes to avoid when working with the derivatives of $f$?

A25: Some common mistakes to avoid when working with the derivatives of $f$ include:

• Failing to check the domain and range of the function.
• Making errors when differentiating the function.
• Failing to consider the concavity of the function.

By understanding the function $f(x) = 4^{-x} - 2$ and its derivatives, you can gain a deeper understanding of the behavior of the function and its applications in real-world scenarios.