Let's solve all the problems step by step:

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Problem 36:

Find $(f+g)(x)$ and $(f-g)(x)$ where $f(x) = 9x^2 + 4$ and $g(x) = 10 - 8x$.

• Solution for $(f+g)(x)$:
  $(f+g)(x) = f(x) + g(x) = (9x^2 + 4) + (10 - 8x)$
  Combine like terms:
  $(f+g)(x) = 9x^2 - 8x + 14$

• Solution for $(f-g)(x)$:
  $(f-g)(x) = f(x) - g(x) = (9x^2 + 4) - (10 - 8x)$
  Distribute the negative and combine like terms:
  $(f-g)(x) = 9x^2 + 8x - 6$

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Problem 37:

Find $(f+g)(x)$ and $(f \cdot g)(x)$ where $f(x) = 9x^2 + 33x + 28$ and $g(x) = 3x + 7$.

• Solution for $(f+g)(x)$:
  $(f+g)(x) = f(x) + g(x) = (9x^2 + 33x + 28) + (3x + 7)$
  Combine like terms:
  $(f+g)(x) = 9x^2 + 36x + 35$

• Solution for $(f \cdot g)(x)$:
  $(f \cdot g)(x) = f(x) \cdot g(x) = (9x^2 + 33x + 28)(3x + 7)$
  Use distribution (FOIL method):
  $f \cdot g = (9x^2)(3x) + (9x^2)(7) + (33x)(3x) + (33x)(7) + (28)(3x) + (28)(7)$
  Simplify each term:
  $f \cdot g = 27x^3 + 63x^2 + 99x^2 + 231x + 84x + 196$
  Combine like terms:
  $(f \cdot g)(x) = 27x^3 + 162x^2 + 315x + 196$

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Problem 38:

Let $f(x) = 81 - x^2$ and $g(x) = 9 - x$. Find $(f+g)(x)$.

• Solution for $(f+g)(x)$:
  $(f+g)(x) = f(x) + g(x) = (81 - x^2) + (9 - x)$
  Combine like terms:
  $(f+g)(x) = -x^2 - x + 90$

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Problem 39:

Given $f(x) = x^2 + 3$ and $g(x) = \frac{x-7}{x}$, find $(g \circ f)(1)$.

• Step 1: Find $f(1)$:
  $f(x) = x^2 + 3 \implies f(1) = 1^2 + 3 = 4$

• Step 2: Find $g(f(1)) = g(4)$:
  $g(x) = \frac{x - 7}{x} \implies g(4) = \frac{4 - 7}{4} = \frac{-3}{4}$

• Final Answer:
  $(g \circ f)(1) = -\frac{3}{4}$

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Problem 40:

Given $f(x) = 7x^2$, find $f^{-1}(x)$ and state whether $f^{-1}(x)$ is a function.

1. Step 1: Solve for $x$ in terms of $y$:
   Let $y = 7x^2$:
   $y = 7x^2 \implies x^2 = \frac{y}{7} \implies x = \pm \sqrt{\frac{y}{7}}$

2. Step 2: Solve for $f^{-1}(x)$:
   Replace $y$ with $x$:
   $f^{-1}(x) = \pm \sqrt{\frac{x}{7}}$

3. Step 3: State whether $f^{-1}(x)$ is a function:
   Since $f^{-1}(x)$ has both a positive and negative square root, it is not a function because it fails the vertical line test.

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Problem 41:

Use symmetry to graph the inverse of the given function.

• Graph Analysis:
  To graph the inverse, reflect the given function across the line $y = x$. The original graph will be symmetric with its inverse about this line.

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Do you want detailed explanations or additional steps for any problem?

Here are 5 related questions to expand your learning:

1. How do you find the composition of two functions, such as $(g \circ f)(x)$?
2. What conditions make the inverse of a function not a function?
3. How can you use symmetry to graph an inverse function?
4. How do you simplify the product of two polynomials?
5. Why is the square root function sometimes not a proper inverse?

Tip: To check if a function has an inverse that is also a function, use the horizontal line test!