Introduction to Relations

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.

A pair belongs to \(R\) only when its first component is in \(A\), its second component is in \(B\), and it satisfies \(y=x+2\). For \(x=1\), \(y=3\), producing \((1,3)\). For \(x=2\), \(y=4\), producing \((2,4)\). For \(x=3\), the equation gives \(y=5\), but 5 is not in \(B\), so that pair is excluded. Hence \(R=\{(1,3),(2,4)\}\), option A.