Answer :
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]
