Question

write an equation for the function in the form $f(x) = a(b)^x$. use whole numbers, decimals, or simplified fractions for the values of a and b. $f(x) = $

Explanation:

Step1: Find the value of \( a \)

We know that the function is in the form \( f(x) = a(b)^x \). When \( x = 0 \), \( f(0) = -1 \) (from the point \( (0, -1) \)). Substitute \( x = 0 \) and \( f(0) = -1 \) into the function:
\( f(0) = a(b)^0 \)
Since \( b^0 = 1 \) for any \( b
eq 0 \), we have \( -1 = a(1) \), so \( a = -1 \).

Step2: Find the value of \( b \)

Now we know \( a = -1 \), and we also have the point \( (1, -7) \). Substitute \( x = 1 \), \( f(1) = -7 \), and \( a = -1 \) into the function \( f(x) = a(b)^x \):
\( -7 = -1(b)^1 \)
Simplify the equation: \( -7 = -b \), so \( b = 7 \).

Step3: Write the function

Now that we have \( a = -1 \) and \( b = 7 \), substitute these values back into the form \( f(x) = a(b)^x \):
\( f(x) = -1(7)^x \) or \( f(x) = -7^x \)

Answer:

\( f(x) = -7^x \) (or \( f(x) = -1 \times 7^x \))