III. DIRECTION: Determine the composite of two functions. (2points each) f(x) = x + 3, g(x)= 2x² - 9 , h(x) = x² 1. f(x) + g(x) 2. f(x) • h(x) 3. g(x) / h(x) 4. g(x)[f(x) + h(x)] 5. h(x) • f(x) – g(x)

Q: If f(x) = (5x – 3)+/3, find f- (x)

A: We have to find inverse function

Q: Evaluate the function for f(x) = x + 3 and g(x) = x2 − 2. (fg)(9)=

A: Given that f(x)=x+3 And g(x)=x2-2 We will find (fg)(9)

Q: Let f(x) = x + 9x and g(x) = 5-x. () Find (x) and (2).

A:

Q: Find f.g, g. f, and g. g. f(x) = x + 4, g(x) = x - 9 (a) f.g (b) g.f

A: Explanation and solutions is given below....

Q: Suppose the functions g and f are defined as follows. h(x) = -5+ 4x2 g (x) = 4 – 9x (ə) Find (4)…

A: Given that, h(x)=-5+4x2 and g(x)=4-9x. Part (a) The value of hg(x) is same as hg(x)=h(x)g(x)…

Q: Consider the function f(x) = 3x° + 2x – 1 and the equation f(x) = 0.

A: Given :-  Consider the function f  x = 3x5 + 2x -1 and the equation f  x  =0.

Q: f(x) = (2-1)³

A: Given: f(x)=2−x415 To sketch the graph of f(x) Let's first find where f(x) will be zero

Q: On which of the following intervals is f(x) = continuous? Vx-2

A:

Q: Use the functions below to evaluate and simplify the function compositions. ƒ₁(x) = 4x² – 3x − 1,…

A: Given : f2(x)=7x2-3x-10   To Find : (f2ο f2)(-1)

Q: Is the function g(x) = x2 one-to-one?

A: Take x=2. g(2)=22=4. Take x=-2. g(-2)=(-2)2=4

Q: Show that the function f (x) = -x3 + 2x2 - 6x + 3 is always decreasing.

A:

Q: f(x) = x* + x³ – 2x² What is the domain of the function given above?

A: We have to find the domain of the given function

Q: .) What is the range of the function y= x^2 - 6? b.) For the function g(x)= x^2 +3x - 5, what is…

A:

Q: Identify the x-intercepts for: f(x) = x² + 5x+6

A: To find x-intercepts for a function f(x),  Put f(x)=0

Q: Use the functions below to evaluate and simplify the function compositions. f₁(x) = 4x² – 3x - 1,…

A:

Q: What is the vertical intercept of the function f (x) = (x + 2)(x − 8)?

A:

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.