To understand why exponential functions never have an x-intercept, we can analyze the properties of exponential functions and their graphs.
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The line \( y=0 \) represents the function's horizontal asymptote, which cannot be crossed. For an x-intercept to exist, \( y \) must equal zero.
This statement is correct. Exponential functions of the form \( y = a \cdot b^x \) (where \( a > 0 \) and \( b > 0 \) with \( b \neq 1 \)) have a horizontal asymptote at \( y = 0 \). The function approaches, but never actually reaches, this line. -
The range of an exponential function is \( y > 0 \), and \( y \) must equal zero for an x-intercept to exist.
This statement is also correct. The range of a standard exponential function is \( (0, \infty) \), which means that the function values are always positive and never equal to zero; therefore, there cannot be any x-intercepts.
So both statements provided are correct, explaining why exponential functions do not have x-intercepts: they approach zero asymptotically and the function values are always positive.
Thus, the correct response is that both initial claims are correct.