Answer :
To determine which exponential functions are equivalent to [tex]\( f(x) = 175(0.80)^{4x} \)[/tex], we need to simplify and compare each of the given functions.
Let's start by simplifying [tex]\( f(x) \)[/tex]:
1. [tex]\( f(x) = 175(0.80)^{4x} \)[/tex]
2. Apply the power rule: [tex]\( (a^b)^c = a^{b \cdot c} \)[/tex].
3. Simplify the exponent: [tex]\( (0.80)^4 = 0.80 \times 0.80 \times 0.80 \times 0.80 = 0.4096 \)[/tex].
4. Thus, [tex]\( f(x) \)[/tex] can be rewritten as [tex]\( f(x) = 175(0.4096)^x \)[/tex].
Now, let's compare this to each of the given functions:
1. [tex]\( w(x) = 175(0.0032)^x \)[/tex]
- [tex]\( 0.0032 \neq 0.4096 \)[/tex]. Not equivalent.
2. [tex]\( q(x) = 175(0.4096)^x \)[/tex]
- Matches [tex]\( f(x) = 175(0.4096)^x \)[/tex]. Equivalent.
3. [tex]\( d(x) = 71.68^x \)[/tex]
- The base [tex]\( 71.68 \neq 0.4096 \)[/tex]. Not equivalent.
4. [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
- Convert: [tex]\( (0.64)^{2x} = ((0.64)^2)^x = (0.4096)^x \)[/tex].
- Simplifies to [tex]\( 175(0.4096)^x \)[/tex]. Equivalent.
5. [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]
- Base [tex]\( 0.1678 \neq 0.4096 \)[/tex]. Not equivalent.
6. [tex]\( u(x) = 175(1-0.20)^x = 175(0.80)^x \)[/tex]
- Matches [tex]\( 175(0.80)^x \)[/tex], but not [tex]\( 175(0.4096)^x \)[/tex]. Not equivalent.
7. [tex]\( h(x) = 175(1-0.5904)^x = 175(0.4096)^x \)[/tex]
- This matches [tex]\( f(x) = 175(0.4096)^x \)[/tex]. Equivalent.
Equivalent functions: [tex]\( q(x) = 175(0.4096)^x \)[/tex], [tex]\( b(x) = 175(0.64)^{2x} \)[/tex], [tex]\( h(x) = 175(1-0.5904)^x \)[/tex].
Let's start by simplifying [tex]\( f(x) \)[/tex]:
1. [tex]\( f(x) = 175(0.80)^{4x} \)[/tex]
2. Apply the power rule: [tex]\( (a^b)^c = a^{b \cdot c} \)[/tex].
3. Simplify the exponent: [tex]\( (0.80)^4 = 0.80 \times 0.80 \times 0.80 \times 0.80 = 0.4096 \)[/tex].
4. Thus, [tex]\( f(x) \)[/tex] can be rewritten as [tex]\( f(x) = 175(0.4096)^x \)[/tex].
Now, let's compare this to each of the given functions:
1. [tex]\( w(x) = 175(0.0032)^x \)[/tex]
- [tex]\( 0.0032 \neq 0.4096 \)[/tex]. Not equivalent.
2. [tex]\( q(x) = 175(0.4096)^x \)[/tex]
- Matches [tex]\( f(x) = 175(0.4096)^x \)[/tex]. Equivalent.
3. [tex]\( d(x) = 71.68^x \)[/tex]
- The base [tex]\( 71.68 \neq 0.4096 \)[/tex]. Not equivalent.
4. [tex]\( b(x) = 175(0.64)^{2x} \)[/tex]
- Convert: [tex]\( (0.64)^{2x} = ((0.64)^2)^x = (0.4096)^x \)[/tex].
- Simplifies to [tex]\( 175(0.4096)^x \)[/tex]. Equivalent.
5. [tex]\( g(x) = 175(0.1678)^{\frac{x}{2}} \)[/tex]
- Base [tex]\( 0.1678 \neq 0.4096 \)[/tex]. Not equivalent.
6. [tex]\( u(x) = 175(1-0.20)^x = 175(0.80)^x \)[/tex]
- Matches [tex]\( 175(0.80)^x \)[/tex], but not [tex]\( 175(0.4096)^x \)[/tex]. Not equivalent.
7. [tex]\( h(x) = 175(1-0.5904)^x = 175(0.4096)^x \)[/tex]
- This matches [tex]\( f(x) = 175(0.4096)^x \)[/tex]. Equivalent.
Equivalent functions: [tex]\( q(x) = 175(0.4096)^x \)[/tex], [tex]\( b(x) = 175(0.64)^{2x} \)[/tex], [tex]\( h(x) = 175(1-0.5904)^x \)[/tex].
