Answer:
The rule that defines the function represented by the table is:
- c. [tex]f(x)=40(\frac{1}{4})^{x}[/tex]
Step-by-step explanation:
To identify the rule that defines the function represented by the table, you must know that [tex]f(x) = y[/tex], so, you can take formula by formula and replace in it the values given in the table, if the result is equal to [tex]y[/tex] replacing the values of [tex]x[/tex], you're gonna know that formula is correct. Let's do that exercise with the formulas in order:
- a. [tex]f(x)=\frac{1}{4}(40)^{x}[/tex]
Now, we're gonna replace the variable [tex]x[/tex] by the first value in the table: 0
- [tex]f(x)=\frac{1}{4}(40)^{0}[/tex]
- [tex]f(x)=\frac{1}{4}*1[/tex]
- [tex]f(x)=\frac{1}{4}[/tex]
How you can see in the table, exactly below the value 0 to the variable [tex]x[/tex] is the value to [tex]y[/tex] (40), how the answer with the A option doesn't result in 40, it doesn't the correct rule.
- b. [tex]f(x)=\frac{1}{2}(40)^{x}[/tex]
We replace with the value 0 again:
- [tex]f(x)=\frac{1}{2}(40)^{0}[/tex]
- [tex]f(x)=\frac{1}{2}*1[/tex]
- [tex]f(x)=\frac{1}{2}[/tex]
In this case, it neither gives us the value 40, for this reason, we pass to the next option:
- c. [tex]f(x)=40(\frac{1}{4})^{x}[/tex]
We replace with the value 0 one more time:
- [tex]f(x)=40(\frac{1}{4})^{0}[/tex]
- [tex]f(x)=40*1[/tex]
- [tex]f(x)=40[/tex]
With this option, the result is equal to the first datum, but, to check this, we can replace the same formula with one or two values more:
- [tex]f(x)=40(\frac{1}{4})^{1}[/tex]
- [tex]f(x)=40*(\frac{1}{4})[/tex]
- [tex]f(x)=10[/tex]
- [tex]f(x)=40(\frac{1}{4})^{2}[/tex]
- [tex]f(x)=40*(\frac{1}{16})[/tex]
- [tex]f(x)=\frac{5}{2}[/tex]
How you can see, with the C option, once you replace the value of [tex]x[/tex], the result is exactly the value for [tex]y[/tex], for this reason, the C option is the correct answer.
