/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Sketch the graph of each functio... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 17

Sketch the graph of each function. $$f(x)=4^{x}$$

Short Answer

Expert verified
The graph of the function \(f(x) = 4^x\) is an upward-sloping curve that passes through the points (-1, 0.25), (0, 1), and (1, 4) and gets infinitely close to, but never touches, the x-axis as \(x\) becomes increasingly negative.

Step by step solution

01

Understand the nature of the function

First, we need to understand the key characteristics of exponential functions. An exponential function is a mathematical function of the following form: \(f(x) = a^x\), where 'a' is a positive constant different from 1. For our function, \(a = 4\). The graph of an exponential function will always pass through the point (0, 1), because any nonzero number raised to the power of 0 is 1.
02

Determine a few key points

Next, we plug in a few values to get some points to plot on the graph. For \(x = -1\), \(f(x) = 4^{-1} = 0.25\). For \(x = 0\), we already know that \(f(x) = 1\), and for \(x = 1\), \(f(x) = 4^1 = 4\). Therefore, three points we can plot on the graph are (-1, 0.25), (0, 1), and (1, 4).
03

Sketch the graph

Plot the points determined above on an x-y graph. Point when \(x = -1\) plots at 0.25 on the y-axis, when \(x = 0\) it plots at 1 on the y-axis and when \(x = 1\) it plots at 4 on the y-axis. In the end, connect the points to show increasing exponential growth.
04

Recognize key functions characteristics

It's crucial to consider that the function has a horizontal asymptote at \(y = 0\). Thus, as \(x\) approaches negative infinity, \(y\) gets infinitely closer to but never reaches zero. Also, as \(x\) increases, \(y\) increases in an exponential pace. This growth is reflected in the way the graph steeply increases for positive x-values.

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Most popular questions from this chapter

Evaluate the given quantity by referring to the function \(f\) given in the following table. $$\begin{array}{cc}x & f(x) \\\\-2 & 1 \\\\-1 & 2 \\\0 & 0 \\\1 & -1 \\\2 & -2\end{array}$$ $$f^{-1}\left(f^{-1}(-2)\right)$$

Solve using any method, and eliminate extraneous solutions. $$e^{\log x}=e$$

Evaluate the expression to four decimal places using a calculator. $$\log 2500$$

Evaluate the expression to four decimal places using a calculator. $$\log 1400$$

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why the equation \(2 e^{x}=-1\) has no solution.
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