Question
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 domain of the function is all real numbers.
The range of the function is all real numbers greater than 3.
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 domain of the function is all real numbers.
The range of the function is all real numbers greater than 3.
Answers
Answered by
GPT-4o mini
Let's analyze the function \( f(x) = 3(2.5)^x \) and check each statement:
1. **The function is exponential.**
**True.** This is an exponential function because it can be expressed in the form \( f(x) = a \cdot b^x \), where \( a = 3 \) and \( b = 2.5 \).
2. **The initial value of the function is 2.5.**
**False.** The initial value (the value of the function when \( x = 0 \)) can be found by evaluating \( f(0) = 3(2.5)^0 = 3(1) = 3 \). Therefore, 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 every increase of 1 in \( x \), the function multiplies by \( 2.5 \). So, \( f(x + 1) = 3(2.5)^{x + 1} = 3(2.5) \cdot (2.5)^x = 2.5 \cdot f(x) \).
4. **The domain of the function is all real numbers.**
**True.** The domain of an exponential function is all real numbers (\( \mathbb{R} \)) since you can input any real number \( x \).
5. **The range of the function is all real numbers greater than 3.**
**False.** The range of the function \( f(x) = 3(2.5)^x \) is all real numbers greater than 0. As \( x \) approaches negative infinity, \( f(x) \) approaches 0, and as \( x \) approaches positive infinity, \( f(x) \) increases without bound.
### Summary of True Statements:
- The function is exponential.
- The function increases by a factor of 2.5 for each unit increase in x.
- The domain of the function is all real numbers.
1. **The function is exponential.**
**True.** This is an exponential function because it can be expressed in the form \( f(x) = a \cdot b^x \), where \( a = 3 \) and \( b = 2.5 \).
2. **The initial value of the function is 2.5.**
**False.** The initial value (the value of the function when \( x = 0 \)) can be found by evaluating \( f(0) = 3(2.5)^0 = 3(1) = 3 \). Therefore, 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 every increase of 1 in \( x \), the function multiplies by \( 2.5 \). So, \( f(x + 1) = 3(2.5)^{x + 1} = 3(2.5) \cdot (2.5)^x = 2.5 \cdot f(x) \).
4. **The domain of the function is all real numbers.**
**True.** The domain of an exponential function is all real numbers (\( \mathbb{R} \)) since you can input any real number \( x \).
5. **The range of the function is all real numbers greater than 3.**
**False.** The range of the function \( f(x) = 3(2.5)^x \) is all real numbers greater than 0. As \( x \) approaches negative infinity, \( f(x) \) approaches 0, and as \( x \) approaches positive infinity, \( f(x) \) increases without bound.
### Summary of True Statements:
- The function is exponential.
- The function increases by a factor of 2.5 for each unit increase in x.
- The domain of the function is all real numbers.