Question
Graph f(x)equals left parenthesis one seventh right parenthesis Superscript x
and g(x)equals log Subscript 1 divided by 7 Baseline x in the same rectangular coordinate system.
Question content area bottom
Part 1
Choose the correct graph below.
A.
-10
10
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10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls steeply in quadrant 2, crosses the vertical axis at 1, and then falls shallowly in quadrant 1 passing through (1, 0.1). As the curve falls to the right it approaches the positive horizontal axis from above. From left to right, a curve falls steeply in quadrant one, crosses the horizontal axis at 1, and then falls at a decreasing rate in quadrant 4 passing through the points (2, negative 0.3) and (3, negative 0.5). As the curve approaches the vertical axis from the right, the curve approaches positive infinity. All coordinates are approximate.
B.
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10
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10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls steeply in quadrant 2, crosses the vertical axis at 1, and then falls shallowly in quadrant 1 passing through (1, 0.1). As the curve falls to the right it approaches the positive horizontal axis from above. From left to right, a curve rises steeply in quadrant 4, crosses the horizontal axis at 1, and then rises at a decreasing rate in quadrant 1 passing through the points (2, 0.4) and (3, 0.6). As the curve approaches the vertical axis from the right, the curve approaches negative infinity. All coordinates are approximate.
C.
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10
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A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls shallowly in quadrant 3, crosses the vertical axis at negative 1, and then falls steeply in quadrant 4 passing through (1, negative 7). As the curve rises to the left it approaches the negative horizontal axis from below. From left to right, a curve falls steeply in quadrant 1, crosses the horizontal axis at 1, and then falls at a decreasing rate in quadrant 4 passing through the points (2, negative 0.3) and (3, negative 0.5). As the curve approaches the vertical axis from the right, the curve approaches positive infinity. All coordinates are approximate.
D.
-10
10
-10
10
and g(x)equals log Subscript 1 divided by 7 Baseline x in the same rectangular coordinate system.
Question content area bottom
Part 1
Choose the correct graph below.
A.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls steeply in quadrant 2, crosses the vertical axis at 1, and then falls shallowly in quadrant 1 passing through (1, 0.1). As the curve falls to the right it approaches the positive horizontal axis from above. From left to right, a curve falls steeply in quadrant one, crosses the horizontal axis at 1, and then falls at a decreasing rate in quadrant 4 passing through the points (2, negative 0.3) and (3, negative 0.5). As the curve approaches the vertical axis from the right, the curve approaches positive infinity. All coordinates are approximate.
B.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls steeply in quadrant 2, crosses the vertical axis at 1, and then falls shallowly in quadrant 1 passing through (1, 0.1). As the curve falls to the right it approaches the positive horizontal axis from above. From left to right, a curve rises steeply in quadrant 4, crosses the horizontal axis at 1, and then rises at a decreasing rate in quadrant 1 passing through the points (2, 0.4) and (3, 0.6). As the curve approaches the vertical axis from the right, the curve approaches negative infinity. All coordinates are approximate.
C.
-10
10
-10
10
A coordinate system has a horizontal axis labeled from negative 10 to 10 in increments of 2 and a vertical axis labeled from negative 10 to 10 in increments of 2. From left to right, a curve falls shallowly in quadrant 3, crosses the vertical axis at negative 1, and then falls steeply in quadrant 4 passing through (1, negative 7). As the curve rises to the left it approaches the negative horizontal axis from below. From left to right, a curve falls steeply in quadrant 1, crosses the horizontal axis at 1, and then falls at a decreasing rate in quadrant 4 passing through the points (2, negative 0.3) and (3, negative 0.5). As the curve approaches the vertical axis from the right, the curve approaches positive infinity. All coordinates are approximate.
D.
-10
10
-10
10
Answers
GPT-4o mini
To choose the correct graph for the functions \( f(x) = \left( \frac{1}{7} \right)^x \) and \( g(x) = \log_{\frac{1}{7}}(x) \), we should analyze the behavior of each function.
1. **Graph of \( f(x) = \left( \frac{1}{7} \right)^x \)**:
- This is an exponential decay function since the base \( \frac{1}{7} < 1 \).
- As \( x \) increases, \( f(x) \) will approach 0 but never actually reach it (horizontal asymptote at \( y = 0 \)).
- At \( x = 0 \), \( f(0) = 1 \).
- For negative \( x \), \( f(x) \) will increase and approach infinity.
2. **Graph of \( g(x) = \log_{\frac{1}{7}}(x) \)**:
- This logarithm function is defined only for \( x > 0 \).
- It is a decreasing function because logarithms with bases less than 1 decrease as \( x \) increases.
- At \( x = 1 \), \( g(1) = 0 \).
- As \( x \) approaches \( 0^+ \), \( g(x) \) will approach infinity.
Now let’s look at how each graph option corresponds to these behaviors:
- **Option A**:
- \( f(x) \) decreases, starting from 1 when \( x = 0 \) and approaching 0 (with correct behavior).
- \( g(x) \) decreases from positive infinity and crosses the x-axis at \( x = 1 \).
- This matches the expected behavior.
- **Option B**:
- \( f(x) \) does match the expected behavior.
- However, \( g(x) \) incorrectly indicates an increase, which is incorrect since it should be decreasing.
- **Option C**:
- The description of \( f(x) \) is incorrect, especially starting from negative values and suggesting it goes to negative infinity.
- **Option D**:
- There is no description provided to analyze.
Based on the analyses, **Option A** is the correct choice since it accurately portrays the behavior of both functions \( f(x) \) and \( g(x) \).
1. **Graph of \( f(x) = \left( \frac{1}{7} \right)^x \)**:
- This is an exponential decay function since the base \( \frac{1}{7} < 1 \).
- As \( x \) increases, \( f(x) \) will approach 0 but never actually reach it (horizontal asymptote at \( y = 0 \)).
- At \( x = 0 \), \( f(0) = 1 \).
- For negative \( x \), \( f(x) \) will increase and approach infinity.
2. **Graph of \( g(x) = \log_{\frac{1}{7}}(x) \)**:
- This logarithm function is defined only for \( x > 0 \).
- It is a decreasing function because logarithms with bases less than 1 decrease as \( x \) increases.
- At \( x = 1 \), \( g(1) = 0 \).
- As \( x \) approaches \( 0^+ \), \( g(x) \) will approach infinity.
Now let’s look at how each graph option corresponds to these behaviors:
- **Option A**:
- \( f(x) \) decreases, starting from 1 when \( x = 0 \) and approaching 0 (with correct behavior).
- \( g(x) \) decreases from positive infinity and crosses the x-axis at \( x = 1 \).
- This matches the expected behavior.
- **Option B**:
- \( f(x) \) does match the expected behavior.
- However, \( g(x) \) incorrectly indicates an increase, which is incorrect since it should be decreasing.
- **Option C**:
- The description of \( f(x) \) is incorrect, especially starting from negative values and suggesting it goes to negative infinity.
- **Option D**:
- There is no description provided to analyze.
Based on the analyses, **Option A** is the correct choice since it accurately portrays the behavior of both functions \( f(x) \) and \( g(x) \).