Question

is ( b ) greater than or less than 2? explain how you know.
write an equation that defines ( g ). show your reasoning.
( f ) and ( g ) represent the number, in thousands, of social media followers as a function of years since 2010. what does the intersection of ( f ) and ( g ) mean?

Explanation:

1. Assume the function \(g(x)\) is an exponential - function of the form \(y = ab^{x}\):

• We know that the function \(g(x)\) passes through the points \((0,3)\) and \((1,12)\).
• Step 1: Find the value of \(a\) using the point \((0,3)\)
• Substitute \(x = 0\) and \(y = 3\) into the equation \(y=ab^{x}\). When \(x = 0\), \(y=ab^{0}=a\). Since \(b^{0}=1\) for \(b

eq0\), we have \(a = 3\). So the equation of the function becomes \(y = 3b^{x}\).

• Step 2: Find the value of \(b\) using the point \((1,12)\)
• Substitute \(x = 1\), \(y = 12\), and \(a = 3\) into the equation \(y = 3b^{x}\). We get \(12=3b^{1}\).
• Solve for \(b\):
• Divide both sides of the equation \(12 = 3b\) by 3. We have \(\frac{12}{3}=b\), so \(b = 4\).

1. Answer the question "Is \(b\) greater than or less than 2?"

• Since \(b = 4\), and \(4>2\), \(b\) is greater than 2.

1. Write the equation of \(g\):

• The equation of the function \(g(x)\) is \(g(x)=3\cdot4^{x}\), because we found that \(a = 3\) and \(b = 4\) from the given points \((0,3)\) and \((1,12)\) on the function \(g\).

Step1: Determine \(a\) using \(x = 0\) point

When \(x = 0\) in \(y=ab^{x}\), \(y=a\). Given \((0,3)\), \(a = 3\).

Step2: Determine \(b\) using \(x = 1\) point

Substitute \(a = 3\), \(x = 1\), \(y = 12\) into \(y = 3b^{x}\), get \(12=3b\), so \(b = 4\).

Step3: Answer the comparison question

Since \(b = 4\), \(b>2\).

Step4: Write the equation of \(g\)

\(g(x)=3\cdot4^{x}\)

Answer:

\(b\) is greater than 2. The equation of \(g\) is \(g(x)=3\cdot4^{x}\)