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We define the following functions: f(x)=2x+5, g(x)=x^2-3, h(x)=7-x/3. Compute (f – h)(4). Evaluate the following two compositions: (fog)(x) and (hog)(x). Transform the g(x) function so that the graph is moved 6 units to the right and 7 units down. Find the inverse functions f^-1(x) and h^-1(x). Write a two to three page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the example and be concise in your reasoning. In the body of your essay, please make sure to include: your solutions to the above problems, making sure to include all mathematical work for both problems, as well as explaining each step.

Paper For Above instruction

Introduction

Mathematics provides a fundamental foundation for understanding functions and their transformations, compositions, and inverses. In this paper, we will analyze specific functions defined as f(x)=2x+5, g(x)=x^2-3, and h(x)=7-x/3. Our tasks include calculating the value of (f – h)(4), evaluating compositions (fog)(x) and (hog)(x), transforming the quadratic function g(x), and finding the inverse functions of f(x) and h(x). These exercises exemplify key concepts in function operations and transformations, which are integral to advanced mathematical understanding.

Part 1: Calculation of (f – h)(4)

First, we need to understand what (f – h)(x) represents. It is the difference between the functions f(x) and h(x):

• f(x) = 2x + 5
• h(x) = 7 - x/3

To find (f – h)(x), we compute:

(f – h)(x) = f(x) - h(x) = (2x + 5) - (7 - x/3) = 2x + 5 - 7 + x/3 = (2x + x/3) - 2

Combine like terms by finding a common denominator for the x terms:

2x = 6x/3, so:

(6x/3 + x/3) - 2 = (7x/3) - 2

This is the general expression, so now evaluate at x=4:

(f – h)(4) = (7 * 4 / 3) - 2 = (28/3) - 2

Expressing 2 as a fraction with denominator 3:

(28/3) - (6/3) = (22/3)

Thus, (f – h)(4) = 22/3 ≈ 7.33

Part 2: Evaluation of Compositions (fog)(x) and (hog)(x)

Next, consider the two compositions:

• (fog)(x) = f(g(x))
• (hog)(x) = h(g(x))

We know g(x) = x^2 - 3. Calculating (fog)(x):

f(g(x)) = f(x^2 - 3) = 2(x^2 - 3) + 5 = 2x^2 - 6 + 5 = 2x^2 - 1

Similarly, for (hog)(x):

h(g(x)) = h(x^2 - 3) = 7 - (x^2 - 3)/3 = 7 - x^2/3 + 1 = (7 + 1) - x^2/3 = 8 - x^2/3

Part 3: Transforming the Function g(x)

We are asked to move the graph of g(x) = x^2 - 3 six units to the right and seven units down.

Horizontal shifts are achieved by replacing x with (x - h), where h is the number of units shifted horizontally. Vertical shifts are done by adding or subtracting the value outside the quadratic.

Apply the transformations step by step:

• Shift 6 units to the right: g(x) becomes g(x - 6) = (x - 6)^2 - 3
• Shift 7 units down: g(x - 6) - 7 = (x - 6)^2 - 3 - 7 = (x - 6)^2 - 10

Therefore, the transformed g(x) function is:

g'(x) = (x - 6)^2 - 10

Part 4: Finding Inverse Functions f⁻¹(x) and h⁻¹(x)

Inverse of f(x) = 2x + 5

To find f⁻¹(x), replace f(x) with y:

y = 2x + 5

Swap variables to solve for x:

x = 2y + 5

Now, solve for y:

2y = x - 5

y = (x - 5)/2

Thus, the inverse function is:

f⁻¹(x) = (x - 5)/2

Inverse of h(x) = 7 - x/3

Set y = 7 - x/3

Swap variables:

x = 7 - y/3

Solve for y:

y/3 = 7 - x

y = 3(7 - x) = 21 - 3x

Rearranged, the inverse function is:

h⁻¹(x) = 21 - 3x

Conclusion

This analysis illustrates fundamental operations involving functions, including difference, composition, transformations, and inverses. Calculations like (f – h)(4), compositions (fog) and (hog), as well as the transformations and inverse functions, deepen our understanding of how functions behave and interact. Such exercises enable students and mathematicians alike to develop a more nuanced grasp of core mathematical concepts, preparing them for more advanced applications in calculus, algebra, and beyond.

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