At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To determine which of the given functions are exponential functions, we need to understand the definition of an exponential function. An exponential function is a function of the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the variable exponent. Let's analyze each function one by one:
(a) \( v(x) = \pi x \)
This function is a linear function because it can be written as \( v(x) = \pi \cdot x \), which is not in the form of \( a^x \). Thus, it is not an exponential function.
(b) \( t(x) = (\sqrt{\pi})^x \)
Here, \( t(x) \) is in the form \( a^x \) where \( a = \sqrt{\pi} \). Since \( \sqrt{\pi} \) is a constant, \( t(x) \) is an exponential function.
(c) \( h(x) = (-\pi)^x \)
This function is of the form \( a^x \) where \( a = -\pi \). Despite \( a \) being negative, it is still a constant raised to the power of \( x \). Therefore, \( h(x) \) is an exponential function.
(d) \( n(x) = \pi^x \)
This function clearly matches the form \( a^x \) where \( a = \pi \), a constant. Thus, \( n(x) \) is an exponential function.
(e) \( p(x) = x^\pi \)
In this case, the base is \( x \) and the exponent is \( \pi \), a constant. This is a power function rather than an exponential function, as it cannot be written as \( a^x \) with a constant base and variable exponent. Therefore, \( p(x) \) is not an exponential function.
Based on this analysis, the functions that are exponential are:
- (b) \( t(x) = (\sqrt{\pi})^x \)
- (c) \( h(x) = (-\pi)^x \)
- (d) \( n(x) = \pi^x \)
Thus, the correct selections for the exponential functions are:
[tex]\[ \boxed{2, 3, 4} \][/tex]
(a) \( v(x) = \pi x \)
This function is a linear function because it can be written as \( v(x) = \pi \cdot x \), which is not in the form of \( a^x \). Thus, it is not an exponential function.
(b) \( t(x) = (\sqrt{\pi})^x \)
Here, \( t(x) \) is in the form \( a^x \) where \( a = \sqrt{\pi} \). Since \( \sqrt{\pi} \) is a constant, \( t(x) \) is an exponential function.
(c) \( h(x) = (-\pi)^x \)
This function is of the form \( a^x \) where \( a = -\pi \). Despite \( a \) being negative, it is still a constant raised to the power of \( x \). Therefore, \( h(x) \) is an exponential function.
(d) \( n(x) = \pi^x \)
This function clearly matches the form \( a^x \) where \( a = \pi \), a constant. Thus, \( n(x) \) is an exponential function.
(e) \( p(x) = x^\pi \)
In this case, the base is \( x \) and the exponent is \( \pi \), a constant. This is a power function rather than an exponential function, as it cannot be written as \( a^x \) with a constant base and variable exponent. Therefore, \( p(x) \) is not an exponential function.
Based on this analysis, the functions that are exponential are:
- (b) \( t(x) = (\sqrt{\pi})^x \)
- (c) \( h(x) = (-\pi)^x \)
- (d) \( n(x) = \pi^x \)
Thus, the correct selections for the exponential functions are:
[tex]\[ \boxed{2, 3, 4} \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.