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 #1
avatar+568 
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To determine the number of values of \( x \) for which \( f(f(x)) = 5 \), we need to analyze the given piecewise function:

 

\[
f(x) = \begin{cases} 
x^2 - 4 & \text{if } x \geq -4 \\
x + 3 & \text{otherwise}
\end{cases}
\]

 

We'll consider each case separately.

 

### Case 1: \( x \geq -4 \)

 

For \( x \geq -4 \), \( f(x) = x^2 - 4 \). We need to find \( y \) such that:

 

\[
f(y) = 5
\]

 

So,

 

\[
y^2 - 4 = 5 \implies y^2 = 9 \implies y = \pm 3
\]

 

However, since \( x \geq -4 \), both solutions \( y = 3 \) and \( y = -3 \) are valid because they are within the domain \( x \geq -4 \). Hence, \( y = 3 \) and \( y = -3 \).

 

Next, we need \( f(x) \) such that:

 

\[
f(x) = 3 \quad \text{or} \quad f(x) = -3
\]

 

#### Subcase 1.1: \( f(x) = 3 \)

 

\[
x^2 - 4 = 3 \implies x^2 = 7 \implies x = \pm \sqrt{7}
\]

 

Since \( x \geq -4 \), both solutions \( x = \sqrt{7} \) and \( x = -\sqrt{7} \) are valid.

 

#### Subcase 1.2: \( f(x) = -3 \)

 

\[
x^2 - 4 = -3 \implies x^2 = 1 \implies x = \pm 1
\]

 

Both solutions \( x = 1 \) and \( x = -1 \) are valid since \( x \geq -4 \).

 

### Case 2: \( x < -4 \)

 

For \( x < -4 \), \( f(x) = x + 3 \). We need to find \( y \) such that:

\[
f(y) = 5
\]

 

So,

 

\[
y + 3 = 5 \implies y = 2
\]

 

However, since \( x < -4 \), the solution \( y = 2 \) does not fall within this domain. Therefore, there are no solutions from this case.

### Conclusion

 

Summarizing the valid solutions from both subcases under \( x \geq -4 \), we have:

 

- \( f(x) = 3 \): \( x = \sqrt{7}, -\sqrt{7} \)
- \( f(x) = -3 \): \( x = 1, -1 \)

 

Thus, we find a total of \( 4 \) values of \( x \):

 

\[
x = \sqrt{7}, -\sqrt{7}, 1, -1
\]

 

Therefore, the number of values of \( x \) for which \( f(f(x)) = 5 \) is \( \boxed{4} \).

09.06.2024
 #1
avatar+568 
0

Here's how to calculate the minimal amount you should bid per bridge for this three-year project:

 

1. Consider the Equipment:

 

Equipment cost: $1,000,000

 

Salvage value (selling price after 3 years): $400,000

 

Depreciation: Straight-line over 3 years

 

Annual depreciation expense: ($1,000,000 - $400,000) / 3 years = $200,000/year

 

2. Factor in Working Capital:

 

Net working capital needed: $250,000 (constant throughout the project)

 

3. Analyze Costs and Taxes:

 

Fixed cost per year: $500,000

 

Variable cost per bridge: $3,000,000

 

Required rate of return: 20%

 

Tax rate: 10%

 

4. Calculate After-Tax Cash Flow per Year:

 

We'll consider one year at a time. Let's denote the year as "Y":

 

Revenue from the bridge (unknown yet - this is what we're solving for): "Bid amount (X)"

 

Total cost: Fixed cost + Variable cost - Depreciation expense

 

Taxable income before depreciation: Bid amount (X) - Total cost

 

Depreciation tax shield: Annual depreciation expense * Tax rate = $200,000 * 10% = $20,000

 

Taxable income: Taxable income before depreciation - Depreciation tax shield

 

Taxes (at 10% rate): Taxable income * Tax rate

 

After-tax cash flow (Y): Bid amount (X) - Total cost - Taxes + Depreciation expense + Net working capital

 

5. Apply Annuity Factor for Present Value:

 

We're considering a three-year project, so we need to find the present value of the after-tax cash flow for each year. Since the cash flow happens at the end of each year (annuity due), we'll use an annuity factor (considering the required rate of return of 20%).

 

Annuity factor for 3 years at 20% interest: 2.106 (given)

 

6. Solve for the Minimum Bid (X):

 

We want the project's Net Present Value (NPV) to be zero at the minimum acceptable bid. NPV is the sum of the present values of all future cash flows.

 

Net Present Value (NPV) Equation:

 

NPV = After-tax cash flow (Year 1) + After-tax cash flow (Year 2) + After-tax cash flow (Year 3) - Initial equipment cost + Salvage value = 0

 

We can rewrite the above equation with the annuity factor and solve for X (the minimum bid amount).

 

After rearranging and substituting the after-tax cash flow formula from step 4:

 

X * (1 - (1 + Required return)^(-Project life)) / Required return = (Total cost - Depreciation expense + Net working capital) * Annuity factor + Equipment cost - Salvage value

 

Plugging in the numbers:

 

X * (1 - (1 + 0.2)^(-3)) / 0.2 = ($500,000 + $3,000,000 - $200,000 + $250,000) * 2.106 + $1,000,000 - $400,000

 

X * (1 - 0.512) / 0.2 = $3,856,000 * 2.106 + $600,000

 

X * 0.488 = $8,038,16 + $600,000

 

X = ($8,038,16 + $600,000) / 0.488 ≈ $8,647,764

 

Minimum Bid amount rounded to nearest $100: $8,647,800

 

Therefore, the minimum amount you should bid per bridge to achieve a zero Net Present Value for this three-year project is approximately $8,647,800.

05.06.2024