To find the equation of the exponential function that passes through the given points [tex]\((0,3)\)[/tex] and [tex]\((3,375)\)[/tex], we will use the general form of an exponential function:
[tex]\[ f(x) = a \cdot b^x \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants.
### Step 1: Use the first point [tex]\((0,3)\)[/tex]
When [tex]\(x = 0\)[/tex], [tex]\(f(x) = 3\)[/tex]:
[tex]\[ f(0) = a \cdot b^0 = a \cdot 1 = a \][/tex]
[tex]\[ a = 3 \][/tex]
### Step 2: Use the second point [tex]\((3,375)\)[/tex]
When [tex]\(x = 3\)[/tex], [tex]\(f(x) = 375\)[/tex]:
[tex]\[ f(3) = 3 \cdot b^3 = 375 \][/tex]
### Step 3: Solve for [tex]\(b\)[/tex]
We now have the equation:
[tex]\[ 3 \cdot b^3 = 375 \][/tex]
Divide both sides by 3:
[tex]\[ b^3 = \frac{375}{3} = 125 \][/tex]
### Step 4: Find the value of [tex]\(b\)[/tex]
Solve for [tex]\(b\)[/tex] by finding the cube root of 125:
[tex]\[ b = \sqrt[3]{125} = 5 \][/tex]
### Step 5: Write the equation
Now that we have the values [tex]\(a = 3\)[/tex] and [tex]\(b = 5\)[/tex], the equation of the exponential function is:
[tex]\[ 3 \cdot 5^x \][/tex]
So, the equation of the exponential function that goes through the points [tex]\((0,3)\)[/tex] and [tex]\((3,375)\)[/tex] is:
[tex]\[ 3 \cdot 5^x \][/tex]
