2 y=f(x) -5 2 (a) Find the domain of the function 1 g(x) = V1- f(x)" Explain your reasoning using complete sentences. Be sure to follow the ap- propriate template. (b) Sketch an accurate graph of the function h(x) = f(\x|). Explain your reasoning completely, using complete sentences. Label the co- ordinates of at least 5 critical points on the graph. %24
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Problem 1
**Use the graph of \( y = f(x) \) shown below to answer the following questions.**
[Insert Graph Image Description Here: The graph of \( y = f(x) \) is plotted on a coordinate plane. The x-axis ranges from -5 to 5, and the y-axis ranges from -4 to 4. The graph appears to be a smooth, continuous curve with a mixture of concave upward and downward sections, having multiple critical points where the slope changes.]
**(a) Find the domain of the function**
\[ g(x) = \frac{1}{\sqrt{1 - f(x)}}. \]
Explain your reasoning using complete sentences. Be sure to follow the appropriate template.
**(b) Sketch an accurate graph of the function**
\[ h(x) = f(|x|). \]
Explain your reasoning completely, using complete sentences. Label the coordinates of at least 5 critical points on the graph.
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### Detailed Graph Description
- **Axes and Scale:**
- The x-axis goes from -5 to 5.
- The y-axis goes from -4 to 4.
- **Graph of \( y = f(x) \):**
- The graph is a continuous curve.
- The curve starts from above the y-axis at an upward slope, then decreases, crosses the y-axis and dips down to a minimum point before rising again, crossing the x-axis, reaching a local maximum, and descending again.
- Key points on the graph include:
- A local minimum around \( (0, -4.4) \).
- A local maximum around \( (2.5, 2.3) \).
- Other key intercepts at the x-axis around \( (1, 0) \), \( (-2.7, 0) \), and at the y-axis \( (0,0) \).
### Questions Breakdown
**(a) Find the domain of the function**
\[ g(x) = \frac{1}{\sqrt{1 - f(x)}}. \]
*Explanation:*
First, we need to determine for which values of \( x \), \( 1 - f(x) \) is positive since the expression inside the square root must be greater than 0 to be defined.
- Find where \( f(x) < 1 \) since \( 1 - f(x) >](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff55b8d36-fa5b-4fad-8629-1f8799139262%2Fe8740a8b-cb89-4e37-9af1-6cd799959272%2Fdmnf46m.png&w=3840&q=75)
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