We need to determine which of the following functions is equivalent to [tex]\( f(x) = 5(1.9)^{3x} \)[/tex]:
1. [tex]\( f(x) = 34.29^x \)[/tex]
2. [tex]\( f(x) = 5(6.86 x) \)[/tex]
3. [tex]\( f(x) = 5(6.86)^x \)[/tex]
4. [tex]\( f(x) = 328.51^x \)[/tex]
To do this, we will examine whether each function can be transformed into the form of [tex]\( f(x) = 5(1.9)^{3x} \)[/tex].
Let’s take a look at each function one by one:
1. [tex]\( f(x) = 34.29^x \)[/tex]
To compare this with [tex]\( f(x) = 5(1.9)^{3x} \)[/tex], observe the base of the exponent. Clearly, [tex]\( 34.29 \)[/tex] is not obviously related to [tex]\( 1.9 \)[/tex]. Therefore, this function does not match.
2. [tex]\( f(x) = 5(6.86 x) \)[/tex]
This function is not an exponential function but rather a linear function multiplied by a constant. The form is not equivalent to [tex]\( 5(1.9)^{3x} \)[/tex].
3. [tex]\( f(x) = 5(6.86)^x \)[/tex]
To see if this function is equivalent, we can compare exponents:
[tex]\[
f(x) = 5(1.9)^{3x}
\][/tex]
Let’s rewrite [tex]\( (1.9)^{3x} \)[/tex] as [tex]\((1.9^3)^x\)[/tex]:
[tex]\[
f(x) = 5(1.9^3)^x
\][/tex]
Now, calculate [tex]\( 1.9^3 \)[/tex]:
[tex]\[
1.9^3 = 1.9 \times 1.9 \times 1.9 \approx 6.859
\][/tex]
Therefore, we get:
[tex]\[
f(x) = 5(6.86)^x
\][/tex]
This matches our given function exactly.
4. [tex]\( f(x) = 328.51^x \)[/tex]
Similar to the first function, the base [tex]\( 328.51 \)[/tex] is not obviously related to [tex]\( 1.9 \)[/tex].
Based on these examinations, the function equivalent to [tex]\( f(x) = 5(1.9)^{3x} \)[/tex] is:
[tex]\( f(x) = 5(6.86)^x \)[/tex]
Therefore, the correct answer is:
[tex]\( f(x) = 5(6.86)^x \)[/tex]
