Find a function of the form `f(x)=a+bc^(x) (c gt 0)`
if `f(0)=15, f(2)=30, f(4)=90`

Answer

To find the function of the form \( f(x) = a + b c^x \) given the conditions \( f(0) = 15 \), \( f(2) = 30 \), and \( f(4) = 90 \), we can follow these steps:

### Step 1: Set up the equations using the given conditions

From the function \( f(x) = a + b c^x \), we can substitute the values of \( x \) to create a system of equations.

1. For \( f(0) = 15 \):
\[ f(0) = a + b c^0 = a + b = 15 \quad \text{(1)} \]

2. For \( f(2) = 30 \):
\[ f(2) = a + b c^2 = 30 \quad \text{(2)} \]

3. For \( f(4) = 90 \):
\[ f(4) = a + b c^4 = 90 \quad \text{(3)} \]

### Step 2: Rearranging the equations

From equation (1):
\[ b = 15 - a \quad \text{(4)} \]

### Step 3: Substitute \( b \) from equation (4) into equations (2) and (3)

Substituting \( b \) into equation (2):
\[ a + (15 - a) c^2 = 30 \]
This simplifies to:
\[ 15 + (15 - a) c^2 - a = 30 \]
\[ (15 - a) c^2 = 15 \quad \text{(5)} \]

Now substituting \( b \) into equation (3):
\[ a + (15 - a) c^4 = 90 \]
This simplifies to:
\[ 15 + (15 - a) c^4 - a = 90 \]
\[ (15 - a) c^4 = 75 \quad \text{(6)} \]

### Step 4: Divide equations (5) and (6)

From equations (5) and (6):
\[ \frac{(15 - a) c^4}{(15 - a) c^2} = \frac{75}{15} \]
Assuming \( 15 - a \neq 0 \), we can simplify:
\[ c^2 = 5 \quad \Rightarrow \quad c = \sqrt{5} \quad \text{(since } c > 0\text{)} \]

### Step 5: Substitute \( c \) back to find \( b \)

Substituting \( c^2 = 5 \) into equation (5):
\[ (15 - a) \cdot 5 = 15 \]
\[ 15 - a = 3 \quad \Rightarrow \quad a = 12 \quad \text{(7)} \]

### Step 6: Find \( b \)

Using equation (4):
\[ b = 15 - a = 15 - 12 = 3 \quad \text{(8)} \]

### Step 7: Write the final function

Now substituting \( a \), \( b \), and \( c \) into the function:
\[ f(x) = 12 + 3(\sqrt{5})^x \]

### Final Answer:
\[ f(x) = 12 + 3 \cdot 5^{x/2} \]