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
9 answers
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).
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
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.
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
Domain: (-∞, -5) ∪ (-5, ∞)
Range: (-∞, 0) ∪ (0, ∞)
The correct description for the domain is: "Domain: (-∞, -5) ∪ (-5, ∞)" and for the range is "Range: (-∞, 0) ∪ (0, ∞)".