Graph each function. ƒ(x)=2−∣x∣ƒ(x)=$$2^{-\left|x\right|}$$

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 38

5장, 문제 38

Graph each function. $ƒ (x) = 2^{- ∣ x ∣}$

검증된 단계별 안내

Identify the function given: $f(x) = 2^{-\lvert x \rvert}$, which involves an exponential expression with a negative absolute value in the exponent.

Understand the role of the absolute value: Since $\lvert x \rvert$ is always non-negative, the exponent $-\lvert x \rvert$ is always zero or negative, meaning the function will produce values between 0 and 1 inclusive.

Create a table of values by choosing several values of $x$ (both positive and negative), calculate $\lvert x \rvert$, then compute $f(x) = 2^{-\lvert x \rvert}$ for each to get points to plot.

Plot the points on the coordinate plane, noting the symmetry about the y-axis because the function depends on $\lvert x \rvert$, which is symmetric.

Sketch the graph by connecting the points smoothly, showing that the function has a maximum at $x=0$ where $f(0) = 2^0 = 1$, and decreases towards zero as $x$ moves away from zero in either direction.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Absolute Value Function

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. In the function f(x) = 2^(-|x|), the absolute value affects the input by making the exponent symmetric about the y-axis, which influences the graph's shape and symmetry.

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is positive and not equal to 1. They exhibit rapid growth or decay depending on the exponent's sign. Here, the negative exponent causes the function to decay as |x| increases, producing a graph that approaches zero but never touches the x-axis.

Graphing Transformations and Symmetry

Graphing involves understanding how transformations like reflections, shifts, and stretches affect the base function. Since the exponent includes -|x|, the graph is symmetric about the y-axis and decreases on both sides from a maximum at x=0. Recognizing this symmetry helps in accurately sketching the function.

Graph each function. ƒ(x)=2−∣x∣ƒ(x)=2^{-\(\left\)|x\(\right\)|}