Explain why the graphs of f(x) =x and g(x) =1/x are in the same quadrants over the same intervals. Does this relationship hold for m(x) = -x and n(x) = -1/x? Does this relationship hold for any function and its reciprocal function.
Well, the graphs of f(x) = x and g(x) = 1/x are indeed in the same quadrants over the same intervals. This is because as x increases, both f(x) and g(x) increase, and as x decreases, both f(x) and g(x) decrease, regardless of their magnitudes. So, they exhibit similar behavior.
Now, when we consider m(x) = -x and n(x) = -1/x, the relationship changes a bit. The graph of m(x) = -x is just the reflection of the graph of f(x) = x across the y-axis. Similarly, the graph of n(x) = -1/x is the reflection of the graph of g(x) = 1/x across the x-axis. Hence, while the intervals and quadrants might remain the same, the orientation of the graphs is flipped.
As for any function and its reciprocal function, the relationship doesn't hold universally. It depends on the specific function and how it interacts with its reciprocal. Some functions may show similar behavior in the same quadrants over the same intervals, while others may not. It's a case-by-case situation, just like deciding whether to eat the last slice of pizza or not – always depends on the particular pizza!
To explain why the graphs of f(x) = x and g(x) = 1/x are in the same quadrants over the same intervals, we need to analyze the behavior of the functions for different values of x.
First, let's consider the function f(x) = x. This is a linear function with a constant slope of 1. As x increases, the value of f(x) also increases, and as x decreases, f(x) decreases. Therefore, f(x) is positive in the right half of the coordinate plane (1st and 2nd quadrants) and negative in the left half (3rd and 4th quadrants).
Now, let's consider the function g(x) = 1/x. This is a reciprocal function, and its behavior is determined by the sign of x. If x is positive, g(x) is also positive, similar to f(x). If x is negative, g(x) is negative. Therefore, g(x) is positive in the 1st and 3rd quadrants and negative in the 2nd and 4th quadrants.
Since both f(x) and g(x) are positive in the 1st quadrant and negative in the 3rd quadrant, and vice versa for the 2nd and 4th quadrants, their graphs are in the same quadrants over the same intervals.
Now let's analyze the functions m(x) = -x and n(x) = -1/x. The function m(x) is a linear function with a slope of -1. Similar to f(x), m(x) is positive in the 4th quadrant and negative in the 2nd quadrant. However, n(x) is different. As x increases, the value of n(x) decreases, and as x decreases, n(x) increases. Therefore, n(x) is negative in the 1st and 3rd quadrants and positive in the 2nd and 4th quadrants.
This means that the relationship mentioned earlier does not hold for m(x) and n(x) since m(x) and n(x) are in different quadrants over the same intervals.
In general, the relationship between any function and its reciprocal function regarding the quadrants they lie in over the same intervals depends on the behavior of the functions themselves. Sometimes they may be in the same quadrants, as with f(x) and g(x), and sometimes they may be in different quadrants, as with m(x) and n(x). It ultimately depends on the shape and behavior of the specific functions.
The relationship between the graphs of f(x) = x and g(x) = 1/x lies in the fact that both functions are positive in the first and third quadrants of the coordinate grid.
To see why this is the case, let's examine each function separately:
1. f(x) = x:
- The graph of f(x) = x is a straight line passing through the origin with a positive slope of 1.
- When x is positive, f(x) is positive, and when x is negative, f(x) is negative.
- Thus, the graph of f(x) is located in the first and third quadrants.
2. g(x) = 1/x:
- The graph of g(x) = 1/x is a hyperbola symmetric about the y-axis with a vertical asymptote at x = 0.
- As x approaches positive infinity, g(x) approaches 0. As x approaches negative infinity, g(x) also approaches 0.
- Therefore, g(x) is positive when x is positive and negative when x is negative.
- This means that the graph of g(x) is also located in the first and third quadrants.
Now, let's consider the functions m(x) = -x and n(x) = -1/x:
1. m(x) = -x:
- The graph of m(x) = -x is a straight line passing through the origin with a negative slope of -1.
- Similar to f(x), when x is positive, m(x) is negative, and when x is negative, m(x) is positive.
- Hence, the graph of m(x) is located in the second and fourth quadrants.
2. n(x) = -1/x:
- The graph of n(x) = -1/x is a hyperbola symmetric about the y-axis with a vertical asymptote at x = 0.
- As x approaches positive infinity or negative infinity, n(x) approaches 0, similar to g(x).
- Consequently, n(x) is positive when x is negative and negative when x is positive.
- Therefore, the graph of n(x) is located in the second and fourth quadrants.
In summary, the relationship where both functions and their reciprocals lie in the same quadrants over the same intervals holds for f(x) = x and g(x) = 1/x, as well as for m(x) = -x and n(x) = -1/x.
This relationship generally holds for any function and its reciprocal function as long as the original function and its reciprocal have the same signs in each respective quadrant.
if x is positive, so is 1/x.
If x is negative, so is 1/x
So, the graphs lie in QI and QIII
apply that logic to m(x) and n(x)