Identify the function defined by y = a*b^x where b > 0 and b ≠ 1.

Identify the function defined by y = a*b^x where b > 0 and b ≠ 1.

• A. Linear Function
• B. Exponential Function
• C. Polynomial Function
• D. Rational Function

Answer: (B)

Recognize that a function where the variable x appears in the exponent is exponential. In this form, the base b is a fixed positive number not equal to 1, and a is a fixed nonzero scalar that scales the graph vertically. The expression b^x grows or decays multiplicatively as x changes, which is the hallmark of exponential behavior, not a linear, polynomial, or rational pattern. If b > 1, the function grows rapidly as x increases; if 0 < b < 1, it decays toward zero as x increases. This distinct shape and behavior come from having the exponent depend on the variable rather than the variable being raised to a fixed power. Hence the function is exponential.

Recognize that a function where the variable x appears in the exponent is exponential. In this form, the base b is a fixed positive number not equal to 1, and a is a fixed nonzero scalar that scales the graph vertically. The expression b^x grows or decays multiplicatively as x changes, which is the hallmark of exponential behavior, not a linear, polynomial, or rational pattern. If b > 1, the function grows rapidly as x increases; if 0 < b < 1, it decays toward zero as x increases. This distinct shape and behavior come from having the exponent depend on the variable rather than the variable being raised to a fixed power. Hence the function is exponential.