The function $f(x)$ is defined as $f(x)=\frac{1}{3}(6)^x$. Which table of values could be used to graph $g(x)$, a reflection of $f(x)$ across the $x$-axis?
\begin{tabular}{|c|c|c|}
\hline$x$ & $f(x)$ & $g(x)$ \\
\hline-2 & $\frac{1}{108}$ & 12 \\
\hline-1 & $\frac{1}{18}$ & 2 \\
\hline 0 & $\frac{1}{3}$ & $\frac{1}{3}$ \\
\hline 1 & 2 & $\frac{1}{18}$ \\
\hline 2 & 12 & $\frac{1}{108}$ \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|}
\hline $x$ & $f x )$ & $g ( x )$ \\
\hline-2 & $\frac{1}{108}$ & 12 \\
\hline-1 & $\frac{1}{18}$ & 2 \\
\hline 0 & 0 & 0 \\
\hline 1 & 2 & $\frac{1}{18}$ \\
\hline 2 & 12 & $\frac{1}{106}$ \\
\hline
\end{tabular}
Answer (2)
Find the expression for g(x) as the reflection of f(x) across the x-axis: g ( x ) = − f ( x ) = − 3 1 ( 6 ) x .
Evaluate g(x) at x = -2, -1, 0, 1, 2: g ( − 2 ) = − 108 1 , g ( − 1 ) = − 18 1 , g ( 0 ) = − 3 1 , g ( 1 ) = − 2 , g ( 2 ) = − 12 .
Compare the calculated values with the given tables to determine the correct one.
Neither table is correct. However, the first table lists the inverse function values instead of the reflection across the x-axis. The second table is also incorrect.
Explanation
Understanding the Problem We are given the function f ( x ) = 3 1 ( 6 ) x and asked to find the table of values for g ( x ) , which is the reflection of f ( x ) across the x -axis. This means that g ( x ) = − f ( x ) . We need to determine which of the two given tables correctly represents the values of g ( x ) .
Finding the Expression for g(x) Since g ( x ) is the reflection of f ( x ) across the x -axis, we have g ( x ) = − f ( x ) . Substituting the expression for f ( x ) , we get g ( x ) = − 3 1 ( 6 ) x .
Evaluating g(x) Now, let's evaluate g ( x ) for the given values of x : -2, -1, 0, 1, and 2.
Calculating g(-2) For x = − 2 , we have g ( − 2 ) = − 3 1 ( 6 ) − 2 = − 3 1 ⋅ 36 1 = − 108 1 .
Calculating g(-1) For x = − 1 , we have g ( − 1 ) = − 3 1 ( 6 ) − 1 = − 3 1 ⋅ 6 1 = − 18 1 .
Calculating g(0) For x = 0 , we have g ( 0 ) = − 3 1 ( 6 ) 0 = − 3 1 ⋅ 1 = − 3 1 .
Calculating g(1) For x = 1 , we have g ( 1 ) = − 3 1 ( 6 ) 1 = − 3 1 ⋅ 6 = − 2 .
Calculating g(2) For x = 2 , we have g ( 2 ) = − 3 1 ( 6 ) 2 = − 3 1 ⋅ 36 = − 12 .
Comparing with the Tables Now, let's compare these calculated values with the values in the given tables.
Analyzing the First Table In the first table, the values for g ( x ) are 12, 2, 3 1 , 18 1 , and 108 1 . These do not match our calculated values of − 108 1 , − 18 1 , − 3 1 , -2, and -12.
Analyzing the First Table (Continued) Let's negate the f ( x ) values in the first table to see if they match the g ( x ) values in the second table. The negated f ( x ) values are − 108 1 , − 18 1 , − 3 1 , -2, and -12. The g ( x ) values in the first table are 12, 2, 3 1 , 18 1 , and 108 1 . Thus, the first table is incorrect.
Analyzing the Tables The correct table should have g ( x ) = − f ( x ) . So, we negate the f ( x ) values in the first table: f ( − 2 ) = 108 1 , f ( − 1 ) = 18 1 , f ( 0 ) = 3 1 , f ( 1 ) = 2 , f ( 2 ) = 12 . Therefore, g ( − 2 ) = − 108 1 , g ( − 1 ) = − 18 1 , g ( 0 ) = − 3 1 , g ( 1 ) = − 2 , g ( 2 ) = − 12 . Neither of the tables match these values. However, the first table lists the inverse function values instead of the reflection across the x-axis. The second table is also incorrect.
Final Analysis However, if we assume there was a typo in the tables and the g ( x ) values in the first table should have been negative, then the first table would be the correct one. The f ( x ) values are 108 1 , 18 1 , 3 1 , 2 , 12 . The reflection across the x-axis would be − 108 1 , − 18 1 , − 3 1 , − 2 , − 12 . If the g ( x ) values in the first table were supposed to be − 12 , − 2 , − 3 1 , − 18 1 , − 108 1 , then the first table would be incorrect.
Conclusion Since g ( x ) = − f ( x ) , we have g ( x ) = − 3 1 ( 6 ) x . Evaluating at x = − 2 , − 1 , 0 , 1 , 2 , we get g ( − 2 ) = − 108 1 , g ( − 1 ) = − 18 1 , g ( 0 ) = − 3 1 , g ( 1 ) = − 2 , g ( 2 ) = − 12 . Neither table matches these values. However, the first table lists the inverse function values instead of the reflection across the x-axis. The second table is also incorrect.
Examples
Reflections across the x-axis are useful in physics when analyzing the trajectory of projectiles. For example, if you model the height of a ball thrown in the air with a function f ( x ) , the reflection g ( x ) = − f ( x ) would represent the inverted trajectory if the ball were somehow 'thrown downwards' with the same force. This concept helps in understanding symmetry and inverse relationships in various physical phenomena. Understanding function transformations such as reflections is also crucial in fields like signal processing, where signals are often manipulated and analyzed using mathematical functions.
The function g ( x ) is the reflection of f ( x ) across the x-axis, defined as g ( x ) = − f ( x ) . Evaluating g ( x ) at specified values gives results that do not match either of the provided tables. Therefore, both tables are incorrect.
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