📝 Expressing functions as compositions (14 MCQs)
📖 From Calculus • 1. Basics before calculus • 14 questions available
What is Expressing functions as compositions?
Definition:
Expressing a function as a composition means decomposing a given function into two or more simpler functions, where the original function equals or , to simplify analysis or computation.
Example:
For , let and , then .
Reason:
This decomposition aids in differentiation using the chain rule, integration by substitution, and understanding step-by-step transformations in complex functions.
📝 All Expressing functions as compositions MCQs
Q1. Which function serves as the inside function \g(x)\ when the function \h(x)=\\sqrt{x+4}\ is written as a composition \h(x)=f(g(x))\?
📖 Explanation: The expression \h(x)=\\sqrt{x+4}\ can be written as \f(g(x))\ with \g(x)\ providing the quantity to be square‑rooted. Here \g(x)=x+4\ supplies the argument of the outer square‑root function \f(u)=\\sqrt{u}\, making option B correct.
Q2. If \h(x)=(3x-2)^{5}\ is expressed as \h(x)=f(g(x))\, which of the following statements about the monotonicity of \f\ and \g\ is necessarily true?
📖 Explanation: The inner function \g(x)=3x-2\ is a linear function with positive slope, so it is strictly increasing. The outer function \f(u)=u^{5}\ is also strictly increasing because odd powers preserve order. Hence both are increasing, giving option A.
Q3. Which decomposition of \(x^{2}+1)^{10}\ yields an inside function that is a polynomial of lower degree than the other options?
📖 Explanation: Choosing \g(x)=x^{2}+1\ gives an inside function of degree 2, which is lower than the degrees in the other choices where the inside function contains higher powers of \(x^{2}+1)\. Thus option C is correct.
Q4. Express \\\sin(x^{3}+2)\ as a composition \f(g(x))\. Which pair \(f,g)\ is correct?
📖 Explanation: To obtain \\\sin(x^{3}+2)\ we first compute \x^{3}+2\ (the inside step) and then apply the sine function. Hence \g(x)=x^{3}+2\ and \f(u)=\\sin u\, which corresponds to option A.
Q5. For \h(x)=\\ln(2x+1)\ written as \h(x)=f(g(x))\, which statement about the domain of the inside function \g\ is necessarily true?
📖 Explanation: The logarithm requires its argument to be positive, so the inside function \g(x)=2x+1\ must satisfy \2x+1>0\. This translates to the domain \x>-\\tfrac{1}{2}\, which is exactly option C.
Q6. Consider \h(x)=e^{(x-1)^{2}}\. Which of the following decompositions makes the outer function linear in its argument?
📖 Explanation: If we let \g(x)=e^{(x-1)^{2}}\ and choose \f(u)=u\ (the identity function), the outer function is linear (of the form \f(u)=u\). This satisfies the requirement, making option D correct.
Q7. Which pair \(f,g)\ correctly represents \h(x)=\\frac{1}{x^{2}+3}\ as a composition \h(x)=f(g(x))\?
📖 Explanation: Writing \h(x)=\\frac{1}{x^{2}+3}\ as \f(g(x))\ suggests letting the inner function produce the denominator, i.e., \g(x)=x^{2}+3\, and the outer function take the reciprocal, \f(u)=\\frac{1}{u}\. This matches option B.
Q8. If \h = f\\circ g\ and both \f\ and \g\ are one‑to‑one functions, which statement must hold for \h\?
📖 Explanation: The composition of two injective (one‑to‑one) functions is itself injective. Since each input maps to a unique output through \g\ and then uniquely through \f\, the overall map \h\ is one‑to‑one, giving option A.
Q9. For \h(x)=\\tan(\\sqrt{x})\, which decomposition correctly identifies the inside function as a root function?
📖 Explanation: The expression \\\tan(\\sqrt{x})\ is evaluated by first taking the square root of \x\ (a root function) and then applying the tangent. Hence \g(x)=\\sqrt{x}\ and \f(u)=\\tan u\, which is option B.
Q10. When differentiating \h(x)=(5x+7)^{4}\ using the chain rule, which inside function simplifies the derivative most effectively?
📖 Explanation: The chain rule works best when the inner function is the linear expression that appears inside the power, i.e., \g(x)=5x+7\. The outer function then becomes \f(u)=u^{4}\, leading to a straightforward derivative; option A captures this.
Q11. Under what condition does swapping the order of composition, i.e., defining \k(x)=g(f(x))\, yield the same function as \h(x)=f(g(x))\?
📖 Explanation: Two functions commute if applying them in either order gives the same result for every input. This is precisely the condition needed for \k(x)=g(f(x))\ to equal \h(x)=f(g(x))\; thus option D is correct.
Q12. Two students decompose \h(x)=\\sqrt{(2x+1)^{2}+4}\. Which decomposition results in the outer function being a square root of a sum?
📖 Explanation: If we let the inside function produce the entire radicand \(2x+1)^{2}+4\, then the outer function is simply the square root of that quantity. This matches option C, where \g(x)=(2x+1)^{2}+4\ and \f(u)=\\sqrt{u}\.
Q13. Which pair \(f,g)\ expresses \h(x)=\\cos(2x-\\pi)\ as a composition \f(g(x))\?
📖 Explanation: The function inside the cosine is \2x-\\pi\. Taking this as the inside function \g(x)=2x-\\pi\ and the outer function as the cosine itself, \f(u)=\\cos u\, reproduces \h(x)\. Hence option B is correct.
Q14. For \h(x)=\\frac{(x^{2}-1)^{3}}{(x-1)^{2}}\, which composition \f(g(x))\ eliminates the removable singularity at \x=1\ by simplifying the expression before applying the outer function?
📖 Explanation: Factor \(x^{2}-1)=(x-1)(x+1)\. Cancelling one \(x-1)\ factor gives \h(x)=(x-1)(x+1)^{3}\. Writing \g(x)=\\frac{x+1}{x-1}\ captures the ratio that removes the singularity, and then \f(u)=u^{3}\ restores the cubic power, matching option A.