Which statements are true of the function $$ f(x)=3(2.5)^{x} $$ ? Check all that apply. The function is exponential. The initial value of the function is 2.5 . The function increases by a factor of 2.5 for each unit increase in $$ x $$. The range of the function is all real numbers greater than 3 .

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Let's evaluate each statement regarding the function $$ f(x) = 3(2.5)^x $$:

1. **The function is exponential.**
   
   - **True.** The function is in the form $$ f(x) = a \cdot b^x $$, which is the standard form of an exponential function.

2. **The initial value of the function is 2.5.**
   
   - **False.** The initial value typically refers to $$ f(0) $$. Calculating $$ f(0) $$:
     $$
     f(0) = 3 \cdot (2.5)^0 = 3 \cdot 1 = 3
     $$
     So, the initial value is **3**, not 2.5.

3. **The function increases by a factor of 2.5 for each unit increase in $$ x $$.**
   
   - **True.** For each increase of 1 in $$ x $$, the function is multiplied by 2.5:
     $$
     f(x+1) = 3 \cdot (2.5)^{x+1} = 3 \cdot (2.5)^x \cdot 2.5 = f(x) \cdot 2.5
     $$

4. **The range of the function is all real numbers greater than 3.**
   
   - **False.** As $$ x $$ approaches negative infinity, $$ (2.5)^x $$ approaches 0, making $$ f(x) $$ approach 0. As $$ x $$ approaches positive infinity, $$ f(x) $$ increases without bound. Therefore, the range is all positive real numbers **greater than 0**, not just greater than 3.

**True Statements:**
- The function is exponential.
- The function increases by a factor of 2.5 for each unit increase in $$ x $$.

Answer: All except statements 2 and 4 are true.
The function is exponential and increases by a factor of 2.5 for each unit increase in $$ x $$.

Which statements are true of the function \( f(x)=3(2.5)^{x} \) ? Check all that apply. The function is exponential. The initial value of the function is 2.5 . The function increases by a factor of 2.5 for each unit increase in \( x \). The range of the function is all real numbers greater than 3 .

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