Answer :
To determine which exponential functions are equivalent to [tex]\( f(x)=175(0.80)^{4x} \)[/tex], we need to simplify the expression and compare it with the given options.
First, let's simplify [tex]\( f(x) \)[/tex]:
1. [tex]\( f(x) = 175 \times (0.80)^{4x} \)[/tex]
2. This is the same as:
[tex]\[ f(x) = 175 \times ((0.80)^4)^x \][/tex]
3. Calculate [tex]\( (0.80)^4 \)[/tex]:
[tex]\[ (0.80)^4 = 0.80 \times 0.80 \times 0.80 \times 0.80 = 0.4096 \][/tex]
4. So, [tex]\( f(x) = 175 \times (0.4096)^x \)[/tex]
Now, let's compare this with the functions provided:
- [tex]\( w(x) = 175(0.0032)^x \)[/tex]: The base [tex]\( 0.0032 \)[/tex] is not equal to [tex]\( 0.4096 \)[/tex].
- [tex]\( q(x) = 175(0.4096)^x \)[/tex]: This is identical to [tex]\( f(x) = 175 \times (0.4096)^x \)[/tex].
- [tex]\( d(x) = 71.68^x \)[/tex]: This does not match the expression for [tex]\( f(x) \)[/tex].
- [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]:
- Simplify [tex]\( (0.64)^{2x} \)[/tex] to [tex]\( ((0.64)^2)^x \)[/tex].
- Calculate [tex]\( (0.64)^2 \)[/tex]:
[tex]\[ (0.64)^2 = 0.64 \times 0.64 = 0.4096 \][/tex]
- So, [tex]\( b(x) = 175 \times (0.4096)^x \)[/tex] which matches [tex]\( f(x) \)[/tex].
- [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]: The base does not simplify to [tex]\( 0.4096 \)[/tex].
- [tex]\( u(x) = 175(1-0.20)^x \)[/tex]: Simplify [tex]\( 1-0.20 \)[/tex]:
[tex]\[ 1-0.20 = 0.80 \][/tex]
- So, [tex]\( u(x) = 175(0.80)^x \)[/tex].
- [tex]\( u(x) \)[/tex] is different because it is [tex]\( (0.80)^x \)[/tex], not [tex]\( (0.80)^{4x} = (0.4096)^x \)[/tex].
- [tex]\( h(x) = 175(1-0.5904)^x \)[/tex]: Simplify [tex]\( 1-0.5904 \)[/tex]:
[tex]\[ 1-0.5904 = 0.4096 \][/tex]
- So, [tex]\( h(x) = 175(0.4096)^x \)[/tex], which matches [tex]\( f(x) \)[/tex].
Based on these calculations, the equivalent functions are:
1. [tex]\( q(x) = 175(0.4096)^x \)[/tex]
2. [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
3. [tex]\( h(x) = 175(1-0.5904)^x \)[/tex]
First, let's simplify [tex]\( f(x) \)[/tex]:
1. [tex]\( f(x) = 175 \times (0.80)^{4x} \)[/tex]
2. This is the same as:
[tex]\[ f(x) = 175 \times ((0.80)^4)^x \][/tex]
3. Calculate [tex]\( (0.80)^4 \)[/tex]:
[tex]\[ (0.80)^4 = 0.80 \times 0.80 \times 0.80 \times 0.80 = 0.4096 \][/tex]
4. So, [tex]\( f(x) = 175 \times (0.4096)^x \)[/tex]
Now, let's compare this with the functions provided:
- [tex]\( w(x) = 175(0.0032)^x \)[/tex]: The base [tex]\( 0.0032 \)[/tex] is not equal to [tex]\( 0.4096 \)[/tex].
- [tex]\( q(x) = 175(0.4096)^x \)[/tex]: This is identical to [tex]\( f(x) = 175 \times (0.4096)^x \)[/tex].
- [tex]\( d(x) = 71.68^x \)[/tex]: This does not match the expression for [tex]\( f(x) \)[/tex].
- [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]:
- Simplify [tex]\( (0.64)^{2x} \)[/tex] to [tex]\( ((0.64)^2)^x \)[/tex].
- Calculate [tex]\( (0.64)^2 \)[/tex]:
[tex]\[ (0.64)^2 = 0.64 \times 0.64 = 0.4096 \][/tex]
- So, [tex]\( b(x) = 175 \times (0.4096)^x \)[/tex] which matches [tex]\( f(x) \)[/tex].
- [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]: The base does not simplify to [tex]\( 0.4096 \)[/tex].
- [tex]\( u(x) = 175(1-0.20)^x \)[/tex]: Simplify [tex]\( 1-0.20 \)[/tex]:
[tex]\[ 1-0.20 = 0.80 \][/tex]
- So, [tex]\( u(x) = 175(0.80)^x \)[/tex].
- [tex]\( u(x) \)[/tex] is different because it is [tex]\( (0.80)^x \)[/tex], not [tex]\( (0.80)^{4x} = (0.4096)^x \)[/tex].
- [tex]\( h(x) = 175(1-0.5904)^x \)[/tex]: Simplify [tex]\( 1-0.5904 \)[/tex]:
[tex]\[ 1-0.5904 = 0.4096 \][/tex]
- So, [tex]\( h(x) = 175(0.4096)^x \)[/tex], which matches [tex]\( f(x) \)[/tex].
Based on these calculations, the equivalent functions are:
1. [tex]\( q(x) = 175(0.4096)^x \)[/tex]
2. [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
3. [tex]\( h(x) = 175(1-0.5904)^x \)[/tex]
