Inverse Functions - Accuplacer Aaf

1. Let $h$ be a one-to-one function such that $h(4) = -7$. What is the value of $h^{-1}(-7)$?
   Answer: $4$ The definition of an inverse function states that if a point $(a, b)$ is on the graph of $h(x)$, meaning $h(a) = b$, then the point $(b, a)$ is on the graph of $h^{-1}(x)$, meaning $h^{-1}(b) = a$. Given $h(4) = -7$, we can identify $a=4$ and $b=-7$. Therefore, $h^{-1}(-7) = 4$.
2. Let $f(x) = x^3 - 7$. What is the value of $f^{-1}(20)$?
   Answer: $3$ To find $f^{-1}(20)$, we need to find the input value $x$ such that the output of $f(x)$ is 20. We set up the equation $f(x) = 20$, which is $x^3 - 7 = 20$. Adding 7 to both sides gives $x^3 = 27$. Taking the cube root of both sides yields $x = 3$.
3. The function $f(x) = (x-2)^2$ is not one-to-one. However, it has an inverse function if its domain is restricted. On which of the following domains does $f(x)$ have an inverse?
   Answer: $[2, \infty)$ A function has an inverse on a domain where it is one-to-one. The graph of $f(x) = (x-2)^2$ is a parabola with its vertex at $x=2$. The function is strictly decreasing for $x<2$ and strictly increasing for $x>2$. To be one-to-one, the domain must be restricted to one side of the vertex. The domain $[2, \infty)$ includes the vertex and all points to its right, where the function is strictly increasing, so it is one-to-one on this domain. On the interval $[1, 3]$, $f(1)=1$ and $f(3)=1$, so it is not one-to-one.
4. The function $h(x) = \frac{2x + 5}{x - 1}$ has domain $x \neq 1$. What is the domain of $h^{-1}(x)$?
   Answer: $x \neq 2$ The domain of $h^{-1}$ equals the range of $h$. To find the range of $h(x) = \frac{2x + 5}{x - 1}$, we solve $y = \frac{2x + 5}{x - 1}$ for $x$: $y(x - 1) = 2x + 5$, so $yx - y = 2x + 5$, giving $x(y - 2) = y + 5$, thus $x = \frac{y + 5}{y - 2}$. This is undefined when $y = 2$, so the range of $h$ is all real numbers except $y = 2$. Therefore, the domain of $h^{-1}$ is $x \neq 2$. Choice A incorrectly uses the domain restriction of $h$. Choices C and D don't reflect the correct range analysis.
5. A one-to-one function $g(x)$ is defined by the set of points $\{(-2, 3), (0, 1), (1, 5), (3, -4)\}$. What is the value of $g^{-1}(3)$?
   Answer: $-2$ To find $g^{-1}(3)$, we are looking for the input to the function $g(x)$ that gives an output of 3. We examine the given points. The point $(-2, 3)$ indicates that $g(-2) = 3$. By the definition of an inverse function, this means $g^{-1}(3) = -2$.
6. If $g(x)$ is a one-to-one function, which statement describes the relationship between the graph of $y = g(x)$ and the graph of $y = g^{-1}(x)$?
   Answer: They are reflections of each other across the line $y = x$. A key property of inverse functions is that their graphs are reflections of each other across the line $y=x$. This is because if a point $(a, b)$ is on the graph of $g(x)$, the point $(b, a)$ is on the graph of $g^{-1}(x)$, and these two points are symmetric with respect to the line $y=x$.
7. If the point $(a, b)$ lies on both the graph of $y = f(x)$ and the graph of $y = f^{-1}(x)$, what must be true about the relationship between $a$ and $b$?
   Answer: $a = b$ When you encounter questions about inverse functions and their graphs, think about the fundamental relationship: if $(a,b)$ is on the graph of $f(x)$, then $(b,a)$ is on the graph of $f^{-1}(x)$. This is because inverse functions "swap" inputs and outputs. If point $(a,b)$ lies on both $y = f(x)$ and $y = f^{-1}(x)$, then we have two conditions simultaneously: - Since $(a,b)$ is on $y = f(x)$: $b = f(a)$ - Since $(a,b)$ is on $y = f^{-1}(x)$: $b = f^{-1}(a)$ From the first condition, if $b = f(a)$, then applying the inverse function to both sides gives us $f^{-1}(b) = a$. But from the second condition, $b = f^{-1}(a)$. Substituting this into our equation: $f^{-1}(f^{-1}(a)) = a$. This is only possible when $a = b$, meaning the point lies on the line $y = x$. Looking at the wrong answers: Choice A ($ab = 1$) would only work for specific functions like $f(x) = \frac{1}{x}$, not generally. Choice B ($a = -b$) would place the point on the line $y = -x$, which isn't the general intersection pattern. Choice D ($a + b = 0$) is equivalent to choice B. Remember this key insight: points that lie on both a function and its inverse must be on the line $y = x$. This line is actually the axis of symmetry for any function and its inverse.
8. Let $f(x)$ be an invertible function. If $f(5) = 1$ and $f(2) = 5$, what is the value of $f^{-1}(f^{-1}(1))$?
   Answer: $2$ This is a two-step problem that must be solved from the inside out. First, we evaluate the inner part, $f^{-1}(1)$. Since we are given $f(5) = 1$, the definition of an inverse tells us that $f^{-1}(1) = 5$. Now we substitute this result back into the expression: we need to find $f^{-1}(5)$. We are given that $f(2) = 5$, so $f^{-1}(5) = 2$. Therefore, $f^{-1}(f^{-1}(1)) = 2$.
9. Let $f(x)$ be an invertible function. What is the value of $f(f^{-1}(10))$?
   Answer: $10$ By the definition of inverse functions, the composition of a function with its inverse yields the input value. That is, $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$. In this case, the input to the composite function is 10, so $f(f^{-1}(10)) = 10$. The specific formula for $f(x)$ is not needed.
10. A function $f(x)$ is defined as $f(x) = \sqrt{x-2}$ for $x \ge 2$. What is the range of the inverse function, $f^{-1}(x)$?
    Answer: $[2, \infty)$ The range of an inverse function $f^{-1}(x)$ is identical to the domain of the original function $f(x)$. The domain of $f(x) = \sqrt{x-2}$ is restricted to values of $x$ for which the expression under the square root is non-negative, which is $x-2 \ge 0$, or $x \ge 2$. This is given as the domain. Therefore, the range of $f^{-1}(x)$ is $[2, \infty)$.
11. Let $h(x) = x^3 + x$. Since $h(x)$ is one-to-one, it has an inverse $h^{-1}(x)$. What is the value of $h^{-1}(h(2))$?
    Answer: $2$ By the definition of inverse functions, $h^{-1}(h(x)) = x$ for all $x$ in the domain of $h$. Here, the input is 2. Therefore, $h^{-1}(h(2)) = 2$. Calculating $h(2) = 2^3 + 2 = 10$ is an intermediate step that is immediately undone by the inverse function.
12. Which of the following functions has an inverse function over its entire domain, $(-\infty, \infty)$?
    Answer: $f(x) = x^3$ A function has an inverse over its domain if and only if it is one-to-one, meaning each output value corresponds to exactly one input value. This can be verified with the horizontal line test. The function $f(x) = x^3$ is strictly increasing, so any horizontal line will intersect its graph only once. The other functions are not one-to-one: $f(x)=x^2$ (e.g., $f(2)=f(-2)=4$), $f(x)=|x|$ (e.g., $f(2)=f(-2)=2$), and $f(x)=5$ (a horizontal line).
13. The function $C(t) = 20t + 50$ gives the total cost in dollars, $C$, to rent a tool for $t$ hours, for $t > 0$. Which of the following best describes the meaning of its inverse function, $C^{-1}(x)$?
    Answer: The number of hours the tool is rented for a total cost of $x$ dollars. The original function $C(t)$ takes hours ($t$) as input and gives total cost ($C$) as output. The inverse function swaps the roles of the input and output. Therefore, $C^{-1}(x)$ must take a total cost (represented by $x$) as input and give the corresponding number of hours as output.
14. The function $f(x) = 2x + 5$ has an inverse $f^{-1}(x)$. What is the value of $3 \cdot f^{-1}(11)$?
    Answer: $9$ First, we must find the value of $f^{-1}(11)$. This is the value of $x$ such that $f(x) = 11$. We set up the equation $2x + 5 = 11$. Subtracting 5 from both sides gives $2x = 6$, and dividing by 2 gives $x = 3$. So, $f^{-1}(11) = 3$. The question asks for $3 \cdot f^{-1}(11)$, which is $3 \cdot 3 = 9$.
15. A one-to-one function $f(x)$ has a domain of $\{x | x \ge 5\}$ and a range of $\{y | y \le -2\}$. What is the domain of its inverse, $f^{-1}(x)$?
    Answer: $\{x | x \le -2\}$ The domain of an inverse function $f^{-1}(x)$ is the range of the original function $f(x)$. The range of $f(x)$ is given as $\{y | y \le -2\}$. Therefore, the domain of $f^{-1}(x)$ is $\{x | x \le -2\}$.
16. If $f$ and $g$ are inverse functions, and $f(3) = 7$, what is the value of $f(g(7))$?
    Answer: $7$ Since $f$ and $g$ are inverse functions, $g = f^{-1}$. Given that $f(3) = 7$, we know that $g(7) = f^{-1}(7) = 3$. Therefore, $f(g(7)) = f(3) = 7$. Alternatively, for any inverse functions, $f(f^{-1}(x)) = x$, so $f(g(7)) = 7$. Choice A gives $g(7)$ instead of $f(g(7))$. Choice C incorrectly adds the values. Choice D suggests insufficient information, but the inverse relationship provides enough information.
17. If $g$ is a one-to-one function and $g^{-1}(4) = -2$, what is the value of $g(-2)$?
    Answer: $4$ Since $g$ is one-to-one, it has an inverse function $g^{-1}$. The key relationship is that if $g^{-1}(a) = b$, then $g(b) = a$. Given that $g^{-1}(4) = -2$, we can conclude that $g(-2) = 4$. Choice A represents $-\frac{1}{g^{-1}(4)}$, confusing inverse function with reciprocal. Choice B represents $\frac{1}{g^{-1}(4)}$, also confusing inverse with reciprocal. Choice C represents $-g^{-1}(4)$, which is the negative of the given value.
18. A function $h(x)$ passes the horizontal line test. If $h(2) = 5$ and $h(7) = -3$, what can be concluded about the inverse function $h^{-1}(x)$?
    Answer: $h^{-1}(5) = 2$ and $h^{-1}(-3) = 7$, but we cannot determine the full domain Since $h$ passes the horizontal line test, it has an inverse function. For inverse functions, if $h(a) = b$, then $h^{-1}(b) = a$. Therefore, $h^{-1}(5) = 2$ and $h^{-1}(-3) = 7$. However, knowing only two points of $h$ doesn't tell us the complete range of $h$, so we cannot determine if $h^{-1}$ exists for all real numbers. Choice A incorrectly assumes the domain of $h^{-1}$ is all reals. Choice B reverses the input-output relationship. Choice D has the wrong coordinate pairs.
19. The graph of an invertible function $f(x)$ contains the point $(3, -5)$. Which of the following points must be on the graph of $f^{-1}(x)$?
    Answer: $(-5, 3)$ If a point $(a, b)$ is on the graph of $f(x)$, this means $f(a) = b$. By the definition of an inverse, $f^{-1}(b) = a$, which means the point $(b, a)$ must be on the graph of $f^{-1}(x)$. Since $(3, -5)$ is on the graph of $f(x)$, the point $(-5, 3)$ must be on the graph of $f^{-1}(x)$.
20. Which of the following functions does NOT have an inverse function?
    Answer: $f(x) = x^2 + 3x + 2$ A function has an inverse if and only if it is one-to-one (passes the horizontal line test). Choice A: $f(x) = |x|$ for $x \geq 0$ is $f(x) = x$ for $x \geq 0$, which is one-to-one on this domain. Choice B: $f(x) = x^3 - 2$ is strictly increasing, so it's one-to-one. Choice C: $f(x) = 2^x$ is strictly increasing, so it's one-to-one. Choice D: $f(x) = x^2 + 3x + 2$ is a parabola opening upward with no domain restriction, so it fails the horizontal line test (for example, $f(-1) = f(-2) = 0$).

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