Introduction to Relations

Step 1: In this set, (a-b) being divisible by (4) occurs only for equal elements. Step 2: So (S) acts like the identity relation, and all its (4) self-pairs also lie in (R). Step 3: Hence (R\cap S) contains (4) pairs.

Step 1: Put (Y=X) on the diagonal. Step 2: Then (X\cap X=X), which is true for every set. Step 3: Therefore every ((X,X)) belongs to the relation, so (R) is reflexive.

Step 1: For reflexivity, put (Y=X). Step 2: Then (X\cup X=X), which is the same as (Y). Step 3: Hence every diagonal pair satisfies the condition.

Step 1: \(\mathcal{P}(S)\) has (8) elements, so (8) diagonal pairs are needed. Step 2: On the diagonal, (X\cup X=X), which equals (S) only when (X=S). Step 3: One diagonal pair is already present, so (7) must be added.

Step 1: On the diagonal, \(X\setminus X=\varnothing\). Step 2: The condition \(X\setminus X=X\) holds only when \(X=\varnothing\). Step 3: Hence only \((\varnothing,\varnothing)\) is a diagonal pair.

Step 1: On the diagonal, (X\triangle X=\varnothing). Step 2: The empty set is a subset of every set, so (\varnothing\subseteq X) is true. Step 3: Therefore all diagonal pairs are in the relation.

Step 1: (f(0)=0+4=4). Step 2: (f(x)-f(0)=(x+4)-4). Step 3: Simplifying gives (x).

Core concept: Evaluate the function at given inputs by substituting the input into the formula. Step 1: \(f(0)=2\cdot0+1=1\). Step 2: \(f(1)=2\cdot1+1=3\). Step 3: Sum is \(1+3=4\), so the correct answer is 4. Why the closest distractor is wrong: option B (3) equals only \(f(1)\), not the sum. Exam tip: Compute each function value separately and then add to avoid mistakes.

Step 1: Substitute the values into the function. \(f(2)=2^2+2=4+2=6\). Step 2: \(f(1)=1^2+2=1+2=3\). Difference: \(6-3=3\). Thus the correct answer is 3. A common wrong choice (4) occurs if someone mis-evaluates \(f(1)\) or forgets the +2 term. Exam tip: always compute each function value separately and then perform the subtraction to avoid sign or arithmetic mistakes.

Substituting \(x=8\) in the function gives \(f(8)=\frac{8-2}{3}=\frac{6}{3}=2\). Hence, the correct answer is 2. The values 3 and 6 result from considering only the denominator or numerator, whereas the complete fraction must be evaluated. Exam tip: To find a function value, substitute the given input directly for \(x\).

Compute values at each input: \(f(10)=10-3=7\) and \(f(4)=4-3=1\). The required difference is \(7-1=6\), so A is correct.
The closest distractor C (7) is wrong because it equals only \(f(10)\), not the difference \(f(10)-f(4)\).
Exam tip: Evaluate the function at each specified x first, then perform subtraction to avoid sign or copying errors.

Concept: \((f+g)(x)=f(x)+g(x)\). So \((f+g)(1)=f(1)+g(1)\). Compute: \(f(1)=1+5=6\) and \(g(1)=1^2=1\), sum = \(6+1=7\). Thus option A is correct. Closest distractor (option B = 6) equals only \(f(1)\) and wrongly omits \(g(1)\). Exam tip: evaluate each function at the given x separately, then add the results.

Here \((fg)(x)\) denotes the product of the two functions, i.e. \((fg)(x)=f(x)\cdot g(x)\). So \(f(1)=1+1=2\) and \(g(1)=1+3=4\). Therefore \((fg)(1)=2\times4=8\). The closest distractor is 6, which arises if someone mistakenly adds the values (\(2+4=6\)) instead of multiplying. Exam tip: remember \((fg)\) means multiply the functions, while \(f\circ g\) denotes composition.

By definition of composition, \((f\circ g)(2)=f(g(2))\). Evaluate the inner function first: \(g(2)=3\cdot2=6\). Then apply \(f\) to that result: \(f(6)=6+2=8\). Hence the correct value is 8. Common mistake: some students stop at \(g(2)=6\) and choose 6, forgetting to apply \(f\); that is incorrect. Exam tip: Always write \((f\circ g)(x)=f(g(x))\) and compute the inner function before the outer one.

The composite function \((h\circ g\circ f)(x)=h(g(f(x)))\), so the functions must be applied from right to left. First, \(f(1)=1+1=2\); then \(g(2)=2\times2=4\); finally, \(h(4)=4^2=16\). Therefore, the correct answer is 16. Option 8 results from reversing the order of \(g\) and \(h\), while 4 is obtained by stopping after the second step. Exam tip: in a function composition, apply the rightmost function first.

In a composition of functions, evaluate the inner function first and then apply the outer function. Thus, \((g\circ f)(4)=g(f(4))\). We have \(f(4)=4-2=2\), so \(g(2)=2^2+1=5\). Therefore, option A is correct. Option B, 17, is obtained by calculating \(g(4)\) directly without first applying \(f\), which is the common error. Exam tip: in \((g\circ f)(x)\), apply \(f\) first and \(g\) second.

Step 1: (f^{-1}(23)) means the value of (x) for which (f(x)=23). Step 2: From (4x+7=23), we get (4x=16), so (x=4). Step 3: To find an inverse value, equate the original function to the given value and solve.

Step 1: Write (f(x+1)=(x+1)^2-4(x+1)). Step 2: Simplifying gives (x^2-2x-3). Step 3: Now (x^2-2x-3-(x^2-4x)=2x-3).

Step 1: The denominator of the function is (x+4). Step 2: The denominator cannot be zero, so (x+4\ne0). Step 3: Removing (x=-4), all other real numbers remain in the domain.

Step 1: Write (x^2-6x+11=(x-3)^2+2). Step 2: Since ((x-3)^2\ge0), (f(x)\ge2). Step 3: The minimum value is (2), so the range is ([2,\infty)).