Question

sec 6.1 exponential functions: proble
(1 point)
are the functions given below exponential functions?

1. $q = t cdot 12^{4}$
2. $q = 2 / 0.2^{t}$
3. $q = 3^{t} + 2^{t}$
4. $q = 12 cdot 4^{t}$
5. $q = 0.5 cdot 3^{0.75t}$
6. $q = 14t^{12}$

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Explanation:

Step1: Recall the form of an exponential function

The general form of an exponential function is \(Q = ab^{t}\), where \(a\) is the initial value (\(a
eq0\)) and \(b>0,b
eq1\) is the base.

Step2: Analyze \(Q = t\cdot12^{4}\)

Here \(12^{4}=20736\), so \(Q = 20736t\). This is a linear function (in the form \(y = mx + c\) with \(c = 0\) and \(m=20736\)), not an exponential function.

Step3: Analyze \(Q=\frac{2}{0.2^{t}}\)

We can rewrite it as \(Q = 2\times(0.2)^{-t}=2\times5^{t}\) (since \(a^{-n}=\frac{1}{a^{n}}\) and \(\frac{1}{0.2}=5\)). It is in the form \(Q = ab^{t}\) (\(a = 2\), \(b = 5\)), so it is an exponential function.

Step4: Analyze \(Q=3^{t}+2^{t}\)

An exponential function is a single - term function of the form \(ab^{t}\). Since \(Q = 3^{t}+2^{t}\) is a sum of two exponential - like terms (not in the form \(ab^{t}\)), it is not an exponential function.

Step5: Analyze \(Q = 12\cdot4^{t}\)

It is in the form \(Q=ab^{t}\) with \(a = 12\) and \(b = 4\), so it is an exponential function.

Step6: Analyze \(Q=0.5\cdot3^{0.75t}\)

Let \(k = 0.75t\), then \(Q=0.5\cdot(3^{0.75})^{t}\). Since \(3^{0.75}=\sqrt[4]{3^{3}}=\sqrt[4]{27}\approx2.2795\), it is in the form \(Q = ab^{t}\) (\(a = 0.5\), \(b=3^{0.75}\)), so it is an exponential function.

Step7: Analyze \(Q = 14t^{12}\)

This is a power function (in the form \(y = ax^{n}\) where \(n = 12\)), not an exponential function.

Answer:

1. No
2. Yes
3. No
4. Yes
5. Yes
6. No