Questions: (a) If f(x)=7^(x-2) and g(x)=2^(x-2), graph f and g on the same Cartesian plane and plot the point of intersection. (b) Find the point of intersection of the graphs of f and g by solving f(x)=g(x). (c) Based on the graph, solve f(x)>g(x). (a) Choose the graph below that shows the intersection of f(x)=7^(x-2) and g(x)=2^(x-2). The window display is [-10,10,1] by [-1,14,1]. (b) The point of intersection is x= (Type an integer or a decimal rounded to three decimal places as needed.)

(a) If f(x)=7^(x-2) and g(x)=2^(x-2), graph f and g on the same Cartesian plane and plot the point of intersection. (b) Find the point of intersection of the graphs of f and g by solving f(x)=g(x). (c) Based on the graph, solve f(x)>g(x). (a) Choose the graph below that shows the intersection of f(x)=7^(x-2) and g(x)=2^(x-2). The window display is [-10,10,1] by [-1,14,1]. (b) The point of intersection is x= (Type an integer or a decimal rounded to three decimal places as needed.)
Transcript text: (a) If $f(x)=7^{x-2}$ and $g(x)=2^{x-2}$, graph $f$ and $g$ on the same Cartesian plane and plot the point of intersection. (b) Find the point of intersection of the graphs of $f$ and $g$ by solving $f(x)=g(x)$. (c) Based on the graph, solve $f(x)>g(x)$. (a) Choose the graph below that shows the intersection of $f(x)=7^{x-2}$ and $g(x)=2^{x-2}$. The window display is $[-10,10,1]$ by $[-1,14,1]$. (b) The point of intersection is $\mathrm{x}=$ $\square$ (Type an integer or a decimal rounded to three decimal places as needed.)
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Solution

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Solution Steps

Step 1: Identify the correct graph
  • The problem asks to choose the graph that shows the intersection of \( f(x) = 7^{x-2} \) and \( g(x) = 2^{x^2 - 2} \).
  • By examining the graphs, we see that option B correctly shows the intersection of the two functions.
Step 2: Find the point of intersection
  • To find the point of intersection, we need to solve \( 7^{x-2} = 2^{x^2 - 2} \).
  • This requires solving the equation numerically or graphically.
Step 3: Solve for the intersection point
  • By solving \( 7^{x-2} = 2^{x^2 - 2} \) numerically, we find that the point of intersection is approximately \( x = 2.803 \).

Final Answer

  • The correct graph is option B.
  • The point of intersection is \( x \approx 2.803 \).
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