Question

question 21 of 25
in general, the point ______ is on the graph of the function $f(x) = a \cdot b^x$.

a. $(a, b)$
b. $(0, 0)$
c. $(b, 0)$
d. $(0, a)$

Explanation:

To determine which point lies on the graph of the function \( f(x) = a \cdot b^x \), we can substitute the \( x \)-coordinate of each option into the function and check if the resulting \( y \)-coordinate matches the given point.

Step 1: Analyze Option A: \((a, b)\)

Substitute \( x = a \) into \( f(x) \):
\( f(a) = a \cdot b^a \).
This is not equal to \( b \) (the \( y \)-coordinate of the point), so \((a, b)\) is not on the graph.

Step 2: Analyze Option B: \((0, 0)\)

Substitute \( x = 0 \) into \( f(x) \):
\( f(0) = a \cdot b^0 \).
Since \( b^0 = 1 \) (for \( b > 0, b
eq 1 \)), this simplifies to \( f(0) = a \cdot 1 = a \).
For this to equal \( 0 \) (the \( y \)-coordinate of the point), we would need \( a = 0 \), but if \( a = 0 \), the function becomes \( f(x) = 0 \), which is a horizontal line (not a typical exponential function). Thus, \((0, 0)\) is not on the graph of a non - trivial exponential function \( f(x)=a\cdot b^x \) (where \( a
eq0 \)).

Step 3: Analyze Option C: \((b, 0)\)

Substitute \( x = b \) into \( f(x) \):
\( f(b) = a \cdot b^b \).
This is not equal to \( 0 \) (the \( y \)-coordinate of the point) unless \( a = 0 \), which again trivializes the function. So \((b, 0)\) is not on the graph.

Step 4: Analyze Option D: \((0, a)\)

Substitute \( x = 0 \) into \( f(x) \):
\( f(0) = a \cdot b^0 \).
Since \( b^0 = 1 \) (for \( b > 0, b
eq 1 \)), this simplifies to \( f(0) = a \cdot 1 = a \).
The \( y \)-coordinate of the point \((0, a)\) is \( a \), which matches \( f(0) = a \). Thus, \((0, a)\) lies on the graph.

Answer:

\(\boldsymbol{D}\)