Identify if the function is exponential growth or decay and justify your response. F(x)=12/17(17/12)^x

To determine if the function $ f(x) = \frac{12}{17} \left( \frac{17}{12} \right)^x $ represents exponential growth or decay, let's examine the base of the exponential term $ \left( \frac{17}{12} \right) $:

1. If $ \left( \frac{17}{12} \right) $ is greater than 1, then the function represents exponential growth.

2. If $ \left( \frac{17}{12} \right) $ is between 0 and 1, exclusive, then the function represents exponential decay.

In this case, $ \left( \frac{17}{12} \right) $ is greater than 1, since 17 is greater than 12. Therefore, the function represents exponential growth.

Justification:

The base of the exponential term $ \left( \frac{17}{12} \right) $ is greater than 1, indicating that the function will increase exponentially as $ x $ increases. This is characteristic of exponential growth, where the function value increases rapidly over time or with increasing input values of $ x $. Hence, the given function $ f(x) $ represents exponential growth.

semsee45 semsee45 · Answer 2 · 2024-02-14T19:27:11+01:00

Answer:

Exponential growth

Step-by-step explanation:

The general form of an exponential function is:

[tex]\boxed{\begin{array}{l}\underline{\textsf{General form of an Exponential Function}}\\\\f(x)=ab^x\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ is the initial value ($y$-intercept).}\\ \phantom{ww}\bullet\;\textsf{$b$ is the base (growth/decay factor) in decimal form.}\end{array}}[/tex]

If a > 0 and b > 1, then the function represents exponential growth.

If a < 0 and b > 1, then the function represents exponential decay.

If a > 0 and 0 < b < 1, then the function represents exponential decay.

If a < 0 and 0 < b < 1, then the function represents exponential growth.

Given exponential function:

[tex]f(x)=\dfrac{12}{17}\left(\dfrac{17}{12}\right)^x[/tex]

In this case:

[tex]a=\dfrac{12}{17} \implies a > 0[/tex]

[tex]b=\dfrac{17}{12} \implies b > 1[/tex]

Therefore, as the initial value (a) is positive and the base (b) is greater than 1, the function represents exponential growth.