Which of the following functions are odd? Select all that apply.
f(x)=x/x^8+8x^4-7
f(x)=x^3+4x
f(x)=3x^5+6x^4+4
f(x)=8x^2-8
Which of the following functions have graphs that contain no asymptotes? Select all that apply.
power; y=x^n
reciprocal; y=1/x
exponetial; y=b^x
logarithmic; y=logb x
root; y=n sqr x
Identify the key characteristics of the parent fifth-root function f(x) 5 sqr x . Include the following: domain, range, intervals where the function is increasing, decreasing, or constant, intercepts, whether the function is even, odd, or neither, whether the function is continuous or discontinuous, asymptotes, and end behavior.
What transformations produce the graph of from the graph of the parent function ? Select all that apply. (1 point)
reflection over the x-axis
reflection over the y-axis
horizontal stretch by a factor of 3
horizontal compression by a factor of 3
vertical stretch by a factor of 3
vertical compression by a factor of 3
When all four transformation types are applied to the same function, which one should be applied third? (1 point)
vertical shift
reflection
horizontal shift
stretch and compression
Describe the transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3. Give the order in which they must be performed to obtain the correct graph.
The functions that are odd are:
- f(x)=x^3+4x
The functions that have graphs that contain no asymptotes are:
- power; y=x^n
- exponential; y=b^x
- logarithmic; y=logb x
The key characteristics of the parent fifth-root function f(x) = 5√x are:
- Domain: All non-negative real numbers [0, ∞)
- Range: All non-negative real numbers [0, ∞)
- Intervals of increase: The function is increasing for all x>0
- Intervals of decrease: The function is decreasing for all x<0
- Intervals of constant: None
- x-intercept: (0,0)
- y-intercept: (0,0)
- Even or Odd: The function is neither even nor odd
- Continuity: The function is continuous for all x in the domain
- Asymptotes: There are no asymptotes
- End behavior: As x approaches infinity, the function approaches infinity. As x approaches negative infinity, the function approaches negative infinity.
The transformations that produce the graph of g(x)=1/2(x-4)^3+5 from the graph of the parent function f(x)=x^3 are:
1. Horizontal shift to the right by 4 units
2. Vertical stretch by a factor of 1/2
3. Vertical shift up by 5 units
The correct order in which the transformations must be performed is:
1. Horizontal shift
2. Vertical stretch
3. Vertical shift
To determine which of the given functions are odd, we need to check if the function satisfies the condition f(-x) = -f(x) for all x in the domain.
For the first function, f(x) = x/x^8+8x^4-7, we can substitute -x into the function and compute f(-x):
f(-x) = (-x) / (-x)^8 + 8(-x)^4 - 7 = -x / x^8 + 8x^4 - 7 = -(x / x^8 + 8x^4 - 7) = -f(x)
Therefore, the first function is odd.
For the second function, f(x) = x^3 + 4x, we can substitute -x into the function and compute f(-x):
f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x
Since -x^3 - 4x is not equal to -f(x) = -(x^3 + 4x), the second function is not odd.
For the third function, f(x) = 3x^5 + 6x^4 + 4, we can substitute -x into the function and compute f(-x):
f(-x) = 3(-x)^5 + 6(-x)^4 + 4 = -3x^5 + 6x^4 + 4
Since -3x^5 + 6x^4 + 4 is not equal to -f(x) = -(3x^5 + 6x^4 + 4), the third function is not odd.
For the fourth function, f(x) = 8x^2 - 8, we can substitute -x into the function and compute f(-x):
f(-x) = 8(-x)^2 - 8 = 8x^2 - 8
Since 8x^2 - 8 is equal to -f(x) = -(8x^2 - 8), the fourth function is odd.
Therefore, the functions that are odd are f(x) = x/x^8+8x^4-7 and f(x) = 8x^2 - 8.
To determine which of the functions have graphs that contain no asymptotes, we need to analyze the behavior of the functions as x approaches positive and negative infinity.
For the power function y = x^n, where n is a real number, there are no asymptotes.
For the reciprocal function y = 1/x, there is an asymptote at x = 0.
For the exponential function y = b^x, where b is a positive real number, there are no asymptotes.
For the logarithmic function y = logb x, where b is a positive number not equal to 1, there is an asymptote at x = 0.
For the root function y = n √ x, where n is an odd positive integer, there are no asymptotes.
Therefore, the functions that have graphs that contain no asymptotes are the power function y = x^n and the exponential function y = b^x.
The key characteristics of the parent fifth-root function f(x) = 5 √ x are as follows:
- Domain: The domain of the function is all real numbers greater than or equal to 0 since the root function is defined only for non-negative numbers.
- Range: The range of the function is all real numbers since the root function can have any real number as an output.
- Intervals of Increasing/Decreasing/Constant: The function is increasing for all positive values of x, constant at x = 0, and decreasing for all negative values of x.
- Intercepts: The x-intercept occurs at (0, 0).
- Even/Odd/Neither: The function is neither even nor odd.
- Continuity/Discontinuity: The function is continuous for all real numbers.
- Asymptotes: There are no asymptotes since the function does not have any vertical or horizontal asymptotes.
- End Behavior: As x approaches positive or negative infinity, the function approaches positive or negative infinity, respectively.
The transformations that produce the graph of g(x) = 1/2(x - 4)^3 + 5 from the parent function f(x) = x^3 are as follows:
1. Horizontal shift to the right by 4 units: (x - 4)
2. Vertical stretch by a factor of 1/2: 1/2(x - 4)
3. Vertical shift upward by 5 units: 1/2(x - 4) + 5
The correct order of the transformations is first the horizontal shift, then the vertical stretch, and finally the vertical shift.
For the first question, to determine if a function is odd, we need to check if f(x) is equal to -f(-x) for all x in the domain. Let's evaluate each of the given functions:
1) f(x) = x / (x^8 + 8x^4 - 7)
2) f(x) = x^3 + 4x
3) f(x) = 3x^5 + 6x^4 + 4
4) f(x) = 8x^2 - 8
To check if a function is odd, we substitute -x in the function and check if we get -f(-x):
1) f(-x) = (-x) / ((-x)^8 + 8(-x)^4 - 7) = -x / (x^8 + 8x^4 - 7) ≠ -f(x)
Therefore, f(x) is not odd.
2) f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x = -f(x)
Therefore, f(x) is odd.
3) f(-x) = 3(-x)^5 + 6(-x)^4 + 4 = -3x^5 + 6x^4 + 4 = -f(x)
Therefore, f(x) is odd.
4) f(-x) = 8(-x)^2 - 8 = 8x^2 - 8 = f(x)
Therefore, f(x) is even.
Therefore, the functions f(x) = x^3 + 4x and f(x) = 3x^5 + 6x^4 + 4 are odd.
For the second question, let's analyze each type of function:
1) Power function; y = x^n
The graph may have asymptotes depending on the value of n.
2) Reciprocal function; y = 1/x
The graph has an asymptote at x = 0.
3) Exponential function; y = b^x
The graph has no asymptotes.
4) Logarithmic function; y = log_b(x)
The graph has an asymptote at x = 0.
5) Root function; y = n√x
The graph has no asymptotes.
Therefore, the power function (y = x^n), reciprocal function (y = 1/x), logarithmic function (y = log_b(x)) have graphs with asymptotes. The exponential function (y = b^x) and root function (y = n√x) have graphs with no asymptotes.
Now let's move on to the next question about the parent fifth-root function.
The parent fifth-root function is f(x) = √[5]{x}. Here are the key characteristics:
- Domain: The domain of the function is all real numbers because any real number can be raised to the 5th root.
- Range: The range is also all real numbers because the function can output any real number when raised to the 5th root.
- Intervals where the function is increasing, decreasing, or constant: The function is always increasing as x increases. It is never constant or decreasing.
- Intercepts: The x-intercept is at (0,0) because any number raised to the 0th power is equal to 1, and the y-intercept is (1,1) because any number raised to the 1st power is equal to itself.
- Even, odd, or neither: The fifth-root function is neither even nor odd because f(-x) ≠ -f(x) and f(-x) ≠ f(x).
- Continuity: The function is continuous for all real numbers.
- Asymptotes: There are no asymptotes for the fifth-root function.
- End Behavior: As x approaches positive or negative infinity, the function approaches positive or negative infinity, respectively.
Now onto the next question about the transformations:
The transformations that produce the graph of g(x) = 1/2(x-4)^3+5 from the parent function f(x) = x^3 are:
1) Horizontal shift of 4 units to the right.
2) Vertical stretch by a factor of 1/2 (compress).
3) Vertical shift 5 units up.
The correct order in which they must be performed is:
1) Horizontal shift of 4 units to the right.
2) Vertical stretch by a factor of 1/2.
3) Vertical shift 5 units up.
Finally, for the last question, when all four transformation types are applied to the same function, the horizontal shift should be applied third. The order of operations for applying the transformations is typically:
1) Vertical shift
2) Vertical stretch or compression
3) Horizontal shift
4) Reflection (if applicable)
Which of the following functions are odd? Select two answers.
f(x)=x/x^8+8x^4-7
f(x)=x^3+4x
f(x)=3x^5+6x^4+4
f(x)=8x^2-8
To determine if a function is odd, we need to check if f(x) = -f(-x) for all x in the domain.
Let's evaluate each of the given functions:
1) f(x) = x/x^8+8x^4-7
To check if this function is odd, we substitute -x into the function:
f(-x) = (-x)/(-x)^8 + 8(-x)^4 - 7 = -x/x^8 + 8x^4 - 7
Since -x/x^8 + 8x^4 - 7 is not equal to -f(x) = -(x/x^8 + 8x^4 - 7), the first function is not odd.
2) f(x) = x^3 + 4x
To check if this function is odd, we substitute -x into the function:
f(-x) = (-x)^3 + 4(-x) = -x^3 - 4x
Since -x^3 - 4x is equal to -f(x) = -(x^3 + 4x), the second function is odd.
3) f(x) = 3x^5 + 6x^4 + 4
To check if this function is odd, we substitute -x into the function:
f(-x) = 3(-x)^5 + 6(-x)^4 + 4 = -3x^5 + 6x^4 + 4
Since -3x^5 + 6x^4 + 4 is equal to -f(x) = -(3x^5 + 6x^4 + 4), the third function is odd.
4) f(x) = 8x^2 - 8
To check if this function is odd, we substitute -x into the function:
f(-x) = 8(-x)^2 - 8 = 8x^2 - 8
Since 8x^2 - 8 is equal to -f(x) = -(8x^2 - 8), the fourth function is odd.
Therefore, the functions f(x) = x^3 + 4x and f(x) = 3x^5 + 6x^4 + 4 are odd.