Question

Graph each function. $$f(x)=10^{-x}$$

   Graph each function.
$$f(x)=10^{-x}$$

Precalculus

Margaret L. Lial,… 4th Edition

Chapter 4, Problem 20

Instant Answer

Solved by Expert Himanshu Kushwaha

03/11/2021

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We can choose a range of x-values and then calculate the corresponding y-values. Show more…

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Graph each function. $$f(x)=10^{-x}$$

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Key Concepts

Growth Behavior

When the base of an exponential function is greater than 1, the function is increasing, meaning that as x increases, the function’s value escalates rapidly. Understanding this growth behavior is critical in anticipating how the graph will rise steeply for positive values of x.

Asymptote

Exponential functions have a horizontal asymptote, usually along the x-axis (y = 0). This asymptote signifies that as x approaches negative infinity, the function's value gets arbitrarily close to zero without ever reaching it. Recognizing asymptotic behavior is essential for accurately graphing the function.

Domain and Range

The domain of an exponential function includes all real numbers, meaning it can accept any real input. Its range is typically limited to positive real numbers because an exponential function will never produce a zero or negative value. This limitation is crucial when graphing, as it defines the set of possible outputs.

Exponential Function

Exponential functions are mathematical functions of the form f(x) = a^x, where a is a positive constant not equal to 1. They are characterized by the variable being in the exponent, leading to either rapid growth (if a > 1) or decay (if 0 < a < 1). This concept is fundamental in understanding phenomena that change multiplicatively over time.

Y-intercept

The y-intercept of an exponential function is found by evaluating the function at x = 0. Because any nonzero number raised to the power of 0 is 1, the y-intercept is conventionally at the point (0, 1). This serves as a key reference point for sketching the graph.

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