Question

assignment 6.1 exponential functions
score: 5.95/10 answered: 7/10
question 8
find the exponential function $f(x) = ab^x$ whose graph goes through the points $(0, 3)$ and $(3, 24)$.
$a = \square$,
$b = \square$,

Explanation:

Step1: Find the value of \(a\)

We know that the function is \(f(x)=ab^{x}\) and it passes through the point \((0, 3)\). Substitute \(x = 0\) and \(f(x)=3\) into the function:
\(3=ab^{0}\)
Since any non - zero number to the power of \(0\) is \(1\) (i.e., \(b^{0}=1\) for \(b
eq0\)), we have \(3 = a\times1\), so \(a = 3\).

Step2: Find the value of \(b\)

Now that we know \(a = 3\), the function becomes \(f(x)=3b^{x}\). The function also passes through the point \((3,24)\). Substitute \(x = 3\), \(f(x)=24\) and \(a = 3\) into the function:
\(24=3b^{3}\)
First, divide both sides of the equation by \(3\):
\(\frac{24}{3}=b^{3}\)
\(8 = b^{3}\)
Taking the cube root of both sides, we get \(b=\sqrt[3]{8}=2\).

Answer:

\(a = 3\), \(b = 2\)