is the following equal? x(t)=((10/9)ce^10t)/(1+(1/9)ce^10t) and x(t)=10/(1+(9e^(-10t)))

Let's examine if the two functions provided are equivalent:

### Function 1:
\[
x(t) = \frac{\frac{10}{9} ce^{10t}}{1 + \frac{1}{9} ce^{10t}}
\]

### Function 2:
\[
x(t) = \frac{10}{1 + 9e^{-10t}}
\]

We will simplify both expressions and check for equivalence.

#### Step 1: Simplifying Function 1
\[
x(t) = \frac{\frac{10}{9} ce^{10t}}{1 + \frac{1}{9} ce^{10t}}
\]
Multiply the numerator and denominator by 9 to eliminate the fractions:
\[
x(t) = \frac{10ce^{10t}}{9 + ce^{10t}}
\]

#### Step 2: Simplifying Function 2
\[
x(t) = \frac{10}{1 + 9e^{-10t}}
\]
Multiply numerator and denominator by \(e^{10t}\):
\[
x(t) = \frac{10e^{10t}}{e^{10t} + 9}
\]

#### Step 3: Comparing the two simplified forms
Function 1:
\[
x(t) = \frac{10ce^{10t}}{9 + ce^{10t}}
\]

Function 2:
\[
x(t) = \frac{10e^{10t}}{9 + e^{10t}}
\]

The expressions look similar, but they will only be equal if \( c = 1 \).

Thus, the two functions are **equivalent only if \( c = 1 \)**.

Would you like further details or have any questions on this? Here are five related questions to consider:

1. How do constants affect the shape of an exponential function?
2. What happens to the behavior of \( x(t) \) as \( t \to \infty \)?
3. Can you manipulate the exponentials in a more general case for different constants?
4. How does simplifying exponentials help in identifying equivalencies?
5. What are some real-world applications of such functions?

**Tip:** Always check if there are hidden constants or factors before concluding that two functions are identical.

Is the following equal? x(t) = ((10/9)ce^10t) / (1 + (1/9)ce^10t) and x(t) = 10 / (1 + 9e^(-10t))

Explore whether the functions x(t) = ((10/9)ce^10t)/(1 + (1/9)ce^10t) and x(t) = 10 / (1 + 9e^{-10t}) are equivalent. This problem involves exponential function simplification and parameter comparison. Suitable for advanced high school or college students.

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