Function composition and inverses - Discrete Math

1. Consider the function $f(x) = \frac{2x + 1}{x - 3}$ where $x \neq 3$. Which of the following statements about $f$ and its inverse $f^{-1}$ is true?
   Answer: $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$ First, let's find $f^{-1}$. Let y = (2x + 1)/(x - 3). Cross-multiplying: y(x - 3) = 2x + 1, so yx - 3y = 2x + 1. Rearranging: yx - 2x = 3y + 1, so x(y - 2) = 3y + 1, giving x = (3y + 1)/(y - 2). Therefore $f^{-1}$(x) = (3x + 1)/(x - 2). The domain of $f^{-1}$ excludes x = 2 (choice A is correct about this), and the range of $f^{-1}$ excludes y = 3 (choice B is also correct about this). For choice C: $f^{-1}$(0) = (3(0) + 1)/(0 - 2) = 1/(-2) = -1/2, and (f ∘ $f^{-1}$)(0) = f($f^{-1}$(0)) = f(-1/2). Now f(-1/2) = (2(-1/2) + 1)/(-1/2 - 3) = (-1 + 1)/(-7/2) = 0/(-7/2) = 0. So choice C is also correct. Choice D states that f($f^{-1}$(x)) = x for all x in the domain of $f^{-1}$, which is the definition of inverse functions and must be true. Let me verify: f($f^{-1}$(x)) = f((3x + 1)/(x - 2)) = (2((3x + 1)/(x - 2)) + 1)/((3x + 1)/(x - 2) - 3). Simplifying the numerator: (2(3x + 1) + (x - 2))/(x - 2) = (6x + 2 + x - 2)/(x - 2) = 7x/(x - 2). Simplifying the denominator: ((3x + 1) - 3(x - 2))/(x - 2) = (3x + 1 - 3x + 6)/(x - 2) = 7/(x - 2). So f($f^{-1}$(x)) = (7x/(x - 2))/(7/(x - 2)) = (7x/(x - 2)) × ((x - 2)/7) = x. This confirms choice D is correct. Since multiple choices appear correct, I need to identify which is the MOST correct or if there's a subtle error in my analysis. Actually, choice D is the fundamental property that must hold, while A, B, and C are specific calculations that could contain errors.
2. Let $f(x) = 3x - 2$ and $g(x) = \frac{x + 2}{3}$. Consider the statements: I) $g = f^{-1}$, II) $(f \circ g)(x) = x$, III) $(g \circ f)(x) = x$. Which statements are true?
   Answer: I, II, and III First verify I: To find $f^{-1}$, let y = 3x - 2, so 3x = y + 2, so x = (y + 2)/3. Thus $f^{-1}$(x) = (x + 2)/3 = g(x), so I is true. For II: (f ∘ g)(x) = f(g(x)) = f((x + 2)/3) = 3((x + 2)/3) - 2 = (x + 2) - 2 = x, so II is true. For III: (g ∘ f)(x) = g(f(x)) = g(3x - 2) = ((3x - 2) + 2)/3 = 3x/3 = x, so III is true. All three statements are correct. Choices A, B, C each miss at least one true statement.
3. Functions $f$ and $g$ satisfy $f(g(x)) = 4x + 3$ and $g(x) = 2x - 1$. If $f^{-1}$ exists, what is $f^{-1}(7)$?
   Answer: $1$ Function composition and inverse problems require you to work systematically through the given relationships. When you see $f(g(x))$ and need to find $f^{-1}$, your goal is first to determine what $f(x)$ looks like. Given $g(x) = 2x - 1$ and $f(g(x)) = 4x + 3$, you can find $f$ by substitution. Since $g(x) = 2x - 1$, you know that $x = \frac{g(x) + 1}{2}$. Substituting this into the composition equation: $f(g(x)) = 4 \cdot \frac{g(x) + 1}{2} + 3 = 2(g(x) + 1) + 3 = 2g(x) + 5$. This means $f(u) = 2u + 5$ (where $u$ represents the input to $f$). To find $f^{-1}(7)$, you need the value that $f$ maps to 7. Set $f(x) = 7$: $2x + 5 = 7$, so $2x = 2$ and $x = 1$. Therefore, $f^{-1}(7) = 1$. Looking at the wrong answers: Choice A ($\frac{3}{2}$) likely comes from incorrectly solving $2x + 3 = 7$ instead of $2x + 5 = 7$. Choice B ($3$) results from solving $2x - 1 = 7$ (confusing $f$ with $g$). Choice D ($\frac{7}{4}$) might arise from setting $4x + 3 = 7$ and solving directly without recognizing that this represents $f(g(x))$, not $f(x)$. Remember: when dealing with function composition, always work step-by-step to isolate the individual functions before finding inverses. Don't try to shortcut by working directly with the composite function.
4. Let $f(x) = x^2 + 1$ and $g(x) = 2x - 3$. If $F(x) = f(g(x))$ and $G(x) = g(f(x))$, what is the value of $F^{-1}(5)$?
   Answer: $\frac{5}{2}$ or $\frac{1}{2}$ First, find F(x) = f(g(x)) = f(2x - 3) = (2x - 3)² + 1 = 4x² - 12x + 9 + 1 = 4x² - 12x + 10. To find $F^{-1}$(5), we need to solve F(x) = 5, which means 4x² - 12x + 10 = 5, so 4x² - 12x + 5 = 0. Using the quadratic formula: x = (12 ± √(144 - 80))/8 = (12 ± √64)/8 = (12 ± 8)/8. This gives x = 20/8 = 5/2 or x = 4/8 = 1/2. Therefore $F^{-1}$(5) = {5/2, 1/2}, which matches choice C. Choice A gives integer values that would result from a different calculation error. Choice B suggests only one solution, missing the fact that quadratics can have two solutions. Choice D would be correct if the discriminant were negative, but 144 - 80 = 64 > 0, so real solutions exist.
5. Functions $p$, $q$, and $r$ satisfy $p(x) = x + 1$, $q(x) = 2x$, and $r(x) = x^2$. What is $((r \circ q) \circ p)(3)$?
   Answer: $64$ When you encounter function composition problems, remember that compositions like $(r \circ q) \circ p$ work from the inside out, just like nested parentheses in algebra. Let's evaluate $((r \circ q) \circ p)(3)$ step by step. First, we need to understand what $(r \circ q)$ means. This creates a new function where we apply $q$ first, then $r$ to that result: $(r \circ q)(x) = r(q(x)) = r(2x) = (2x)^2 = 4x^2$. Now we can evaluate $((r \circ q) \circ p)(3)$. This means we apply $p$ first to get $p(3) = 3 + 1 = 4$, then apply our composite function $(r \circ q)$ to that result: $(r \circ q)(4) = 4(4)^2 = 4 \cdot 16 = 64$. Looking at the wrong answers: Choice A ($49$) likely comes from incorrectly computing $7^2$ after making an error in the composition order. Choice B ($36$) suggests someone calculated $6^2$, possibly from adding instead of following the composition properly. Choice D ($16$) results from computing just $4^2$, which would be correct if you were only applying $r$ to $p(3)$ but ignores the $q$ function entirely. The correct answer is C: $64$. Study tip: Always work function compositions from right to left (or inside to outside). Write out each step explicitly rather than trying to do multiple compositions mentally—this prevents order-of-operations errors that create most wrong answers in composition problems.
6. Let $f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}$ and $g(x) = x + 1$. What is $(f \circ g)(1.5)$?
   Answer: $4$ To find (f ∘ g)(1.5), first compute g(1.5) = 1.5 + 1 = 2.5. Then compute f(2.5). Since 2.5 > 1, use the second piece: f(2.5) = 2(2.5) - 1 = 5 - 1 = 4. Choice A incorrectly uses f(1.5) instead of f(g(1.5)). Choice C incorrectly applies the first piece f(x) = x² to get (2.5)² = 6.25. Choice D incorrectly computes 2(2.5) - 1 as 4.25.
7. Functions $f$ and $g$ are defined by $f(x) = x^2 + 1$ for $x \geq 0$ and $g(x) = \sqrt{x - 1}$ for $x \geq 1$. What is the domain of $(g \circ f)(x)$?
   Answer: $x \geq 0$ For (g ∘ f)(x) = g(f(x)) to be defined, we need: (1) x in domain of f, so x ≥ 0, and (2) f(x) in domain of g, so f(x) ≥ 1. Since f(x) = x² + 1 and x ≥ 0, we have f(x) ≥ 1 for all x ≥ 0. Therefore, the domain is x ≥ 0. Choice B assumes we need f(x) ≥ 1 but ignores that this is automatically satisfied. Choice C incorrectly solves x² + 1 = 2. Choice D ignores domain restrictions entirely.
8. Consider functions $f: A \rightarrow B$ and $g: B \rightarrow C$ where $f(x) = x + 2$, $g(x) = 3x - 1$, $A = \{1, 2, 3\}$, $B = \{3, 4, 5\}$, and $C = \{8, 11, 14\}$. Which statement about the composition $g \circ f$ is true?
   Answer: $g \circ f$ is well-defined and equals $\{(1,8), (2,11), (3,14)\}$ First check that f maps A into B: f(1) = 3, f(2) = 4, f(3) = 5. All outputs are in B, so f is well-defined. Now compute (g ∘ f): (g ∘ f)(1) = g(f(1)) = g(3) = 3(3) - 1 = 8; (g ∘ f)(2) = g(4) = 11; (g ∘ f)(3) = g(5) = 14. So g ∘ f = {(1,8), (2,11), (3,14)}. Choice B incorrectly computes g ∘ f directly as 3(x+2)-1 without checking domains. Choice C incorrectly claims f(3) ∉ B when f(3) = 5 ∈ B. Choice D gives the wrong domain.
9. Let $f(x) = 2x - 1$ and $g(x) = \frac{x + 3}{2}$. If $h(x) = (f \circ g)(x)$, which of the following statements about $h$ and its inverse is correct?
   Answer: $h(x) = x + 2$ and $h^{-1}(x) = x - 2$ First, find h(x) = (f ∘ g)(x) = f(g(x)) = f((x+3)/2) = 2((x+3)/2) - 1 = (x+3) - 1 = x + 2. To find the inverse, let y = x + 2, then x = y - 2, so $h^{-1}$(x) = x - 2. Choice B incorrectly assumes h is its own inverse. Choice C has the functions reversed. Choice D incorrectly computes the composition as 2x + 2.

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