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Chapter 3: Problem 79

Explain why the functions \(F(x)=1.4^{x}\) and \(G(x)=e^{0.336 x}\) represent essentially the same function.

Short Answer

Expert verified
From the steps taken above, it's clear that the two functions \(F(x) = 1.4^x\) and \(G(x) = e^{0.336x}\) represent the same function due to properties of exponential functions and logarithms.

Step by step solution

01

Write both functions

The functions given are \(F(x) = 1.4^x\) and \(G(x) = e^{0.336x}\)
02

Write function F in terms of e

Every exponential function can be written in terms of \(e\). So, we can write the function \(F(x)\) as \(F(x) = e^{x \cdot \ln 1.4}\)
03

Compare 1.4 to e

For \(F(x)\), we have \(\ln 1.4\), and for \(G(x)\), we have \(0.336\). We want to look at these expressions closely. Compute \(\ln 1.4\) and notice it roughly equals \(0.336\)
04

Rewrite F(x) and compare

Now that we know \(\ln 1.4 \approx 0.336\), we can write \(F(x) = e^{\ln 1.4 \cdot x} = e^{0.336 x}\). So, it's clear that \(F(x)\) and \(G(x)\) represent the same function.

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