Graph ​f(x)equals 4 Superscript x and ​g(x)equals log Subscript 4 Baseline x in the same rectangular coordinate system.

Question

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Part 1
Choose the correct graph below.
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A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls shallowly in quadrant 3, crosses the y-axis at negative 1, and then falls steeply in quadrant 4 passing through (1, negative 4). As the curve rises to the left it approaches the negative x-axis from below. From left to right, a curve falls steeply in quadrant 1, crosses the x-axis at 1, and then falls at a decreasing rate in quadrant 4 passing through the points (2, negative 0.5) and (3, negative 0.8). As the curve approaches the y-axis from the right, the curve approaches positive infinity. All coordinates are approximate.
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A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve rises shallowly in quadrant 2, crosses the y-axis at 1, and then rises steeply in quadrant 1 passing through (1, 4). As the curve falls to the left it approaches the negative x-axis from above. From left to right, a curve rises shallowly in quadrant 3, crosses the y-axis at negative 3, and then rises steeply passing through (1, 0). As the curve falls to the left it approaches the y-value negative 4 from above. All coordinates are approximate.
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A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve rises shallowly in quadrant 2, crosses the y-axis at 1, and then rises steeply in quadrant 1 passing through (1, 4). As the curve falls to the left it approaches the negative x-axis from above. From left to right, a curve rises steeply in quadrant four, crosses the x-axis at 1, and then rises at a decreasing rate in quadrant 1 passing through the points (2, 0.5) and (3, 0.8). As the curve approaches the y-axis from the right, the curve approaches negative infinity. All coordinates are approximate.
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Answer 1

To graph the functions \( f(x) = 4^x \) and \( g(x) = \log_4(x) \), let's analyze their characteristics:

Function: \( f(x) = 4^x \)

1. Behavior:

   • The function \( f(x) \) is an exponential function.
   • As \( x \) approaches negative infinity, \( f(x) \) approaches 0. (The curve approaches the x-axis but never touches it).
   • As \( x \) increases, \( 4^x \) rapidly increases, crossing the y-axis at \( (0, 1) \) because \( f(0) = 4^0 = 1 \).

2. Key points:

   • \( f(-1) = 4^{-1} = \frac{1}{4} \)
   • \( f(1) = 4^1 = 4 \)

Function: \( g(x) = \log_4(x) \)

1. Behavior:

   • The function \( g(x) \) is a logarithmic function.
   • It is only defined for \( x > 0 \) (passes through the y-axis where \( x=1 \)).
   • As \( x \) approaches 0 from the right, \( g(x) \) approaches negative infinity.
   • As \( x \) increases, \( g(x) \) increases, crossing the x-axis at \( (1, 0) \) because \( g(1) = \log_4(1) = 0 \).

2. Key points:

   • \( g(4) = \log_4(4) = 1 \)
   • \( g(16) = \log_4(16) = 2 \)

Summary of Graph Characteristics

To identify the correct graph based on the analysis:

• The graph of \( f(x) = 4^x \) should rise steeply in quadrant 1 and go through \( (0, 1) \) and \( (1, 4) \).
• The graph of \( g(x) = \log_4(x) \) should start from negative infinity when \( x \) approaches 0, cross the x-axis at \( (1, 0) \) and rise through points like \( (4, 1) \).

Looking at Options

• The second option seems to describe these behaviors correctly, stating that from left to right, the curve of \( f(x) \) rises steeply in quadrant 1, and from left to right, the curve of \( g(x) \) rises in quadrant 3, crossing the y-axis at \( (1, 0) \) and approaching negative infinity as it gets close to the y-axis from the right.

So, the correct graph is the second option.