Which of the following functions are one-to-one? Select all that apply.
a) lower f left parenthesis x right parenthesis equals x superscript 3 baseline minus 7Image with alt text: lower f left parenthesis x right parenthesis equals x superscript 3 baseline minus 7
b) lower f left parenthesis x right parenthesis equals x superscript 2 baseline minus 4
Image with alt text: lower f left parenthesis x right parenthesis equals x superscript 2 baseline minus 4
c) lower f left parenthesis x right parenthesis equals Start Fraction 1 over 8 x minus 1 End Fraction
Image with alt text: lower f left parenthesis x right parenthesis equals Start Fraction 1 over 8 x minus 1 End Fraction
d) lower f left parenthesis x right parenthesis equals Start Fraction 5 over x superscript 4 baseline End Fraction
Image with alt text: lower f left parenthesis x right parenthesis equals Start Fraction 5 over x superscript 4 baseline End Fraction
e) lower f left parenthesis x right parenthesis equals Start Absolute Value x End Absolute Value
To determine if a function is one-to-one, we need to check if every value in the domain corresponds to a unique value in the range. In other words, if two different inputs produce the same output, the function is not one-to-one.
a) The function f(x) = x^3 - 7 is one-to-one because no two different inputs will produce the same output.
b) The function f(x) = x^2 - 4 is not one-to-one because if we take x = 2 and x = -2, both will produce the same output f(2) = f(-2) = 0.
c) The function f(x) = (1/8)x - 1 is one-to-one because no two different inputs will produce the same output.
d) The function f(x) = 5/x^4 is one-to-one because no two different inputs will produce the same output.
e) The function f(x) = |x| is not one-to-one because both x and -x will produce the same output f(x) = f(-x).
Therefore, the functions that are one-to-one are a), c), and d).
only 2 answers are allowed
In that case, the two functions that are one-to-one are a) and c).
Find the inverse function for lower f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis superscript 3 baseline minus 2.
To find the inverse function for f(x) = (x + 1)^3 - 2, we need to swap the roles of x and f(x) and solve for the new f(x), which will be the inverse function.
Let y = (x + 1)^3 - 2
Swap x and y:
x = (y + 1)^3 - 2
Now, solve for y:
x = (y + 1)^3 - 2
x + 2 = (y + 1)^3
∛(x + 2) = y + 1
∛(x + 2) - 1 = y
Therefore, the inverse function of f(x) = (x + 1)^3 - 2 is:
f^(-1)(x) = ∛(x + 2) - 1
Determine the domain and range for the inverse of lower f left parenthesis x right parenthesis equals Start Fraction 1 over x End Fraction plus 5.
To determine the domain and range of the inverse function of f(x) = 1/x + 5, we need to consider the restrictions and possible values for the input and output.
First, let's find the inverse function of f(x):
y = 1/x + 5
Swap x and y:
x = 1/y + 5
Solve for y:
x - 5 = 1/y
1/y = x - 5
Invert both sides:
y = 1/(x - 5)
The inverse function of f(x) is:
f^(-1)(x) = 1/(x - 5)
Now, let's determine the domain and range of f^(-1)(x):
Domain of f^(-1)(x) is the set of all possible values for x that result in a meaningful output. In this case, the denominator (x - 5) cannot be zero, since division by zero is undefined. Therefore, the domain is:
Domain: x ≠ 5
Range of f^(-1)(x) is the set of all possible values for y (or x in this case) that result from the function. Since the function is always positive (due to 1 divided by a positive value), the range is:
Range: y > 0
Therefore, the domain of the inverse function is x ≠ 5 and the range is y > 0.
choices
domain: left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma infinity right parenthesis
range: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma infinity right parenthesis range: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
domain: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
range: left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis range: Image with alt text: left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma infinity right parenthesis
domain: left parenthesis negative infinity comma 5 right parenthesis union left parenthesis 5 comma infinity right parenthesis
range: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma 5 right parenthesis union left parenthesis 5 comma infinity right parenthesis range: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
domain: left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis
range: left parenthesis negative infinity comma 5 right parenthesis union left parenthesis 5 comma infinity right parenthesis
The correct answer is:
Domain: (-∞, -5) ∪ (-5, ∞)
Range: (-∞, 0) ∪ (0, ∞)
The correct description for the domain is: "Domain: (-∞, -5) ∪ (-5, ∞)" and for the range is "Range: (-∞, 0) ∪ (0, ∞)".
To determine which functions are one-to-one, we need to check if each input value has a unique output value. If any input value produces the same output value, then the function is not one-to-one.
a) The function f(x) = x^3 - 7:
To check if this function is one-to-one, we need to define two input values, say x1 and x2, which are different but give the same output value. By substituting these values into the function and comparing the results, we can determine if it is one-to-one.
b) The function f(x) = x^2 - 4:
Similar to the previous case, we can check if this function is one-to-one by comparing the outputs for different input values.
c) The function f(x) = (1/8)x - 1:
Again, we need to check if different input values yield the same output value.
d) The function f(x) = 5/(x^4):
We apply the same process of checking different inputs and examining the corresponding outputs.
e) The function f(x) = |x| (absolute value of x):
Since absolute value always returns non-negative results, this function is not one-to-one. For example, both input values -2 and 2 will give the same output, 2.
By comparing the outputs for different input values in each function, we can identify which ones are one-to-one.