📝 Graphing inverse functions reflection (13 MCQs)

📖 Explanation: The domain of an inverse function consists exactly of the values that the original function outputs. Since the range of $f$ is the set of all possible outputs, those values become the permissible inputs for $f^{-1}$. Therefore the domain of $f^{-1}$ equals the range of $f$.

📖 Explanation: First compute the inner function: $f(3)=2\cdot3=6$. Then apply the inverse: $f^{-1}(6)=\frac{1}{2}\cdot6=3$. The composition $f^{-1}(f(x))$ always returns the original input $x$. Hence the correct result is 3, which appears as option C.

📖 Explanation: An inverse function is obtained by reflecting the original graph across the line $y=x$. For $f(x)=x^{3}$ and its inverse, the two curves are mirror images about that line. The statement that captures this relationship is that they are symmetric about the line $y=x$.

📖 Explanation: By definition of an inverse, the ordered pair $(a,b)$ on $g$ becomes $(b,a)$ on $g^{-1}$. Swapping the coordinates of $(4,9)$ yields $(9,4)$. This point must lie on the inverse’s graph, making option B the correct choice.

📖 Explanation: The equation $h(0)=5$ tells us that the input 0 maps to the output 5. The inverse function reverses this mapping, sending 5 back to the original input. Hence $h^{-1}(5)=0$, which is listed as option D.

📖 Explanation: The domain of an inverse function is precisely the range of the original function. Since $p$ outputs values in $[0,7]$, those become the allowable inputs for $p^{-1}$. Therefore the domain of $p^{-1}$ is $[0,7]$, which corresponds to option C.

📖 Explanation: The function $q(x)=1/x$ is its own inverse, meaning its graph is unchanged when reflected across the line $y=x$. This property implies symmetry about that line. The graph does not coincide with the line, nor is it orthogonal; the correct description is symmetry about $y=x$.

📖 Explanation: Because $r$ is invertible, the inverse function undoes the original mapping. The statement $r(2)=-3$ tells us that input 2 yields output $-3$. Applying the inverse to $-3$ returns the original input, so $r^{-1}(-3)=2$, which is option A.

📖 Explanation: The derivative of an inverse function at a point equals the reciprocal of the original function’s derivative at the corresponding point, provided both derivatives exist. Here $\frac{1}{2}$ and 2 are reciprocal numbers ($\frac{1}{2}\times2=1$). Hence the slopes are reciprocals, making option B correct.

📖 Explanation: For an inverse to exist, the original function must be one‑to‑one. On the interval $(- \infty,0]$ the rule $x+1$ maps to $(- \infty,1]$; on $(0,\infty)$ the rule $2x$ maps to $(0,\infty)$. The overlap $(0,1]$ means two different inputs produce the same output, violating injectivity. Hence no inverse exists, option B.

📖 Explanation: First compute $g(b)=f^{-1}(b)+a$. Since $f(a)=b$, the inverse satisfies $f^{-1}(b)=a$. Substituting gives $g(b)=a+a=2a$. Therefore the value of $g(b)$ is twice $a$, which matches option D.

📖 Explanation: The composition $f\circ f^{-1}$ applied to any element of the domain of $f^{-1}$ (which is the range of $f$) returns that element unchanged, because the inverse undoes the original function. Since $q$ is in the range of $f$, $(f\circ f^{-1})(q)=q$. Option A is correct.

📖 Explanation: The graph of an inverse function is obtained by reflecting the original graph across the line $y=x$. This mirror operation swaps each point $(x,y)$ with $(y,x)$. No translation, rotation, or reflection across the x‑axis achieves this effect, so option C accurately describes the relationship.