Answer :
Sure! Let's determine which exponential functions are equivalent to [tex]\( f(x) = 175(0.80)^{4x} \)[/tex].
An exponential function is of the form [tex]\( a(b^x) \)[/tex], where [tex]\( a \)[/tex] is the initial value and [tex]\( b \)[/tex] is the base. To find equivalent functions, we need to see if the expressions inside these given functions can be rewritten to match the form of [tex]\( f(x) \)[/tex].
1. [tex]\( w(x) = 175(0.0032)^x \)[/tex]
This function is not equivalent. The base 0.0032 doesn't come from any manipulation of 0.80 raised to a power related to 4x.
2. [tex]\( q(x) = 175(0.4096)^x \)[/tex]
This is indeed equivalent because [tex]\( 0.4096 \)[/tex] is [tex]\( (0.80)^4 \)[/tex]. Therefore, [tex]\( (0.80)^{4x} = (0.4096)^x \)[/tex].
3. [tex]\( d(x) = 71.68^x \)[/tex]
This function is not equivalent. The base doesn't relate to the manipulation of the base 0.80.
4. [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
This is equivalent. Here, [tex]\( 0.64 = (0.80)^2 \)[/tex]. Therefore, [tex]\( (0.80)^{4x} \)[/tex] can be rewritten as [tex]\( ((0.80)^2)^{2x} = (0.64)^{2x} \)[/tex].
5. [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]
This is equivalent. If we take [tex]\( (0.80)^{4x} \)[/tex] and express it as [tex]\( (0.80)^{2 \cdot 2x} = ((0.80)^2)^{2x} \)[/tex], upon further simplifying it using exponents, we get a similar form as [tex]\( 0.1678 \)[/tex].
6. [tex]\( u(x) = 175(1-0.20)^x \)[/tex]
This is equivalent because [tex]\( 1 - 0.20 = 0.80 \)[/tex], and although it's not raised to 4x, considering just [tex]\( 0.80 \)[/tex], they match without the 4x exponent.
7. [tex]\( h(x) = 175(1-0.5904)^x \)[/tex]
* This function is not equivalent. The subtraction results in a number that doesn't simplify to 0.80 raised to the needed power setups.
In conclusion, the equivalent functions to [tex]\( f(x) = 175(0.80)^{4x} \)[/tex] are:
- [tex]\( q(x) = 175(0.4096)^x \)[/tex]
- [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
- [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]
- [tex]\( u(x) = 175(1-0.20)^x \)[/tex]
These correspond to options 2, 4, 5, and 6.
An exponential function is of the form [tex]\( a(b^x) \)[/tex], where [tex]\( a \)[/tex] is the initial value and [tex]\( b \)[/tex] is the base. To find equivalent functions, we need to see if the expressions inside these given functions can be rewritten to match the form of [tex]\( f(x) \)[/tex].
1. [tex]\( w(x) = 175(0.0032)^x \)[/tex]
This function is not equivalent. The base 0.0032 doesn't come from any manipulation of 0.80 raised to a power related to 4x.
2. [tex]\( q(x) = 175(0.4096)^x \)[/tex]
This is indeed equivalent because [tex]\( 0.4096 \)[/tex] is [tex]\( (0.80)^4 \)[/tex]. Therefore, [tex]\( (0.80)^{4x} = (0.4096)^x \)[/tex].
3. [tex]\( d(x) = 71.68^x \)[/tex]
This function is not equivalent. The base doesn't relate to the manipulation of the base 0.80.
4. [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
This is equivalent. Here, [tex]\( 0.64 = (0.80)^2 \)[/tex]. Therefore, [tex]\( (0.80)^{4x} \)[/tex] can be rewritten as [tex]\( ((0.80)^2)^{2x} = (0.64)^{2x} \)[/tex].
5. [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]
This is equivalent. If we take [tex]\( (0.80)^{4x} \)[/tex] and express it as [tex]\( (0.80)^{2 \cdot 2x} = ((0.80)^2)^{2x} \)[/tex], upon further simplifying it using exponents, we get a similar form as [tex]\( 0.1678 \)[/tex].
6. [tex]\( u(x) = 175(1-0.20)^x \)[/tex]
This is equivalent because [tex]\( 1 - 0.20 = 0.80 \)[/tex], and although it's not raised to 4x, considering just [tex]\( 0.80 \)[/tex], they match without the 4x exponent.
7. [tex]\( h(x) = 175(1-0.5904)^x \)[/tex]
* This function is not equivalent. The subtraction results in a number that doesn't simplify to 0.80 raised to the needed power setups.
In conclusion, the equivalent functions to [tex]\( f(x) = 175(0.80)^{4x} \)[/tex] are:
- [tex]\( q(x) = 175(0.4096)^x \)[/tex]
- [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
- [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]
- [tex]\( u(x) = 175(1-0.20)^x \)[/tex]
These correspond to options 2, 4, 5, and 6.
