Algebra of real functions

Correct answer: A. (4x+2)

In addition, like (x) terms are added, so ((f+g)(x)=x+2+3x=4x+2). In exams, open brackets first.

Correct answer: A. 3x-5

The difference of two functions is defined as (f-g)(x)=f(x)-g(x). Thus, (5x-1)-(2x+4)=5x-1-2x-4=3x-5. Option C has the wrong sign for the constant term because the signs of all terms of the second function change during subtraction. Exam tip: enclose the second function in brackets before subtracting.

Correct answer: A. \(x^2-1\)

For the product of two real functions, \((fg)(x)=f(x)g(x)\). Therefore, \((fg)(x)=(x+1)(x-1)=x^2-1\), using the identity \(a^2-b^2=(a+b)(a-b)\). Option C, \(2x\), results from adding the functions, not multiplying them. In an exam, distinguish the product \((fg)(x)\) from the composition \((f\circ g)(x)\).

Correct answer: A. \(x+3\)

By the definition of the quotient of two functions, \(\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}\). Hence, \(\frac{2x+6}{2}=x+3\). Option B results from dividing only the constant term by 2, which is incorrect. Exam tip: divide the entire numerator by the denominator and check that the denominator is non-zero; here \(g(x)=2\), so every real \(x\) is allowed.

Correct answer: A. 8

The sum of two functions is defined by \((f+g)(x)=f(x)+g(x)\). Hence, \((f+g)(2)=f(2)+g(2)=2^2+4=4+4=8\). Option B gives only \(f(2)\) and does not include \(g(2)\). Exam tip: For \((f+g)(a)\), find the values of both functions at \(a\) and add them.

Correct answer: A. -3

\((f-g)(1)=f(1)-g(1)\). Here, \(f(1)=3\) and \(g(1)=1+5=6\), so \((f-g)(1)=3-6=-3\). Option B results from reversing the subtraction order. Exam tip: In \((f-g)(a)\), always subtract the value of the second function from that of the first.

Correct answer: A. 12

(fg)(1)=f(1)×g(1)=(1+2)(1+3)=3×4=12, so the correct answer is 12. The value 7 results from adding f(1) and g(1), but the question asks for their product. Exam tip: Rewrite (fg)(x) as f(x)g(x) before substituting the given value.

Correct answer: A. 5

The governing concept is the quotient of two real-valued functions. By definition, (f/g)(x) = f(x)/g(x), provided that g(x) is not zero. Evaluate the numerator at x = 3: f(3) = 3^2 − 4 = 9 − 4 = 5. Evaluate the denominator separately: g(3) = 3 − 2 = 1. Therefore (f/g)(3) = f(3)/g(3) = 5/1 = 5, so option A is correct. The denominator is non-zero, so the quotient is defined. Option B is only the denominator value, option C is the unadjusted square 3^2, and option D is not obtained from the quotient rule. Separating numerator and denominator prevents confusion between function operations and ordinary addition or subtraction.

Correct answer: A. x² + 2x + 1

The sum of two functions is defined as \((f+g)(x)=f(x)+g(x)\). Therefore, \((f+g)(x)=(x²+1)+2x=x²+2x+1\), so option A is correct. Option B incorrectly changes the sign of 2x. Exam tip: In function addition, add the function values at the same input x and then combine like terms.

Correct answer: A. \(3x+10\)

The difference of two functions is defined by \((f-g)(x)=f(x)-g(x)\). Thus, \((f-g)(x)=(4x+7)-(x-3)=4x+7-x+3=3x+10\). In option C, the sign of the constant term has been handled incorrectly. Exam tip: when subtracting a function, change the signs of every term inside its bracket.

Correct answer: A. \(2x^3\)

For the product of two functions, (fg)(x)=f(x)g(x). Hence, (fg)(x)=(2x)(x^2)=2x^3. Option B, 2x^2, represents f(g(x)), the composition of the functions, not their product. Exam tip: distinguish (fg)(x) from f(g(x)).

Correct answer: A. (2x)

\(\left(\frac{f}{g}\right)(x)=\frac{6x^2}{3x}=2x\), because \(x\neq0\). In quotient questions, check the condition first.

Correct answer: A. 2x+8

Multiplying a function f by 2 gives (2f)(x)=2f(x). Hence, (2f)(x)=2(x+4)=2x+8. Option C is incorrect because it multiplies only the x-term, not the constant term 4. Exam tip: Distribute a scalar multiplier to every term of the function.

Correct answer: A. \(-3x+2\)

\((-f)(x)=-f(x)=-(3x-2)=-3x+2\). The negative sign applies to the entire function, so the signs of both terms change. Option B changes only the sign of the constant term and is therefore incorrect. Exam tip: when a minus sign precedes a bracket, change the sign of every term inside it.

Correct answer: A. \\(2x^2\\)

The sum of two functions is found pointwise: \\((f+f)(x)=f(x)+f(x)=x^2+x^2=2x^2\\). Hence, option A is correct. Option B results from incorrectly multiplying the terms, but the operation here is addition. Exam tip: when adding like terms, add their coefficients and keep the common power unchanged.

Correct answer: A. \(x^2+2x+1\)

For the product of functions, \((ff)(x)=f(x)\cdot f(x)=(x+1)(x+1)=(x+1)^2=x^2+2x+1\). Hence, option A is correct. Option B, \(f(x)+f(x)=2x+2\), represents the sum of the functions, not their product. In exams, carefully distinguish between \(ff\) and \(f+f\).

Correct answer: A. 0

The governing concept is pointwise subtraction of functions. For every value of x in the common domain, (f − g)(x) is defined as f(x) − g(x). Since the two given functions are identical, substitute their expressions: (f − g)(x) = (x + 5) − (x + 5). Removing the parentheses gives x + 5 − x − 5, and like terms cancel to produce 0. Thus the difference is the zero function and option A is correct. Option B would be obtained by adding the functions, option C merely repeats one original function, and option D has no basis in the subtraction. The result is zero for every permissible x, not just for one particular input.

Correct answer: B. \(4x^2+5x-1\)

\((f+g)(x)=f(x)+g(x)\). So \((f+g)(x)=(2x^2+3)+(2x^2+5x-1)= (2x^2+2x^2)+5x+(3-1)=4x^2+5x-1\).

Correct answer: A. x² − 2x

The governing operation is multiplication of real-valued functions, defined by (fg)(x) = f(x)g(x). Substituting the given functions gives (fg)(x) = 1·(x² − 2x) = x² − 2x. The constant function 1 acts as the multiplicative identity, so it leaves g(x) unchanged for every x in the common domain. Option B would result from adding 1 instead of multiplying; option C confuses the product with f alone; and option D changes the sign of the linear term without justification. Since both functions are defined for every real x, there is no additional domain restriction. Therefore option A is correct.

Correct answer: A. \(2x\)

By the definition of the sum of two functions, \((f+g)(x)=f(x)+g(x)\). Therefore, \((f+g)(x)=(x-2)+(x+2)=2x\), because the constant terms \(-2\) and \(+2\) cancel each other. Option B corresponds to the product \(f(x)g(x)\), while option C considers only the constant terms. Exam tip: In \(f+g\), add the function values; do not multiply them.

Correct answer: A. 3x

The difference of two functions is (f − g)(x) = f(x) − g(x). Substituting the given expressions gives x² + 2x − (x² − x) = x² + 2x − x² + x = 3x. The minus sign changes the sign of each term in g(x).

Correct answer: A. 2

For the product of two functions, \((fg)(x)=f(x)g(x)\). Thus, \(f(0)=2\) and \(g(0)=1\), so \((fg)(0)=2\times1=2\). Option D is only \(g(0)\), while option C represents the sum of the two values. Exam tip: interpret \((fg)(x)\) as the product of functions, not their composition.

Correct answer: A. 4

For the difference of two functions, \((f-g)(2)=f(2)-g(2)\). Here, \(f(2)=2^2+3=7\) and \(g(2)=2^2-1=3\), so \((f-g)(2)=7-3=4\). The value 8 results from adding the two function values, so it is incorrect. Exam tip: For \((f-g)(a)\), evaluate both functions at \(a\) and then subtract.

Correct answer: C. −6

The governing rule for adding functions is (f+g)(x)=f(x)+g(x). We first evaluate both functions at the required input x=−1. Substitution gives f(−1)=2(−1)+1=−2+1=−1. Similarly, g(−1)=−1−4=−5. Adding the two results yields (f+g)(−1)=−1+(−5)=−6. Therefore option C is correct. An equivalent check is to add the functions first: (f+g)(x)=(2x+1)+(x−4)=3x−3, and then substitute x=−1 to obtain −3−3=−6. Option A can arise from a sign or arithmetic error, option B ignores the negative input, and option D does not follow from either evaluation. Careful substitution preserves the signs.

Correct answer: A. \(\frac{5}{2}\)

\(\left(\frac{f}{g}\right)(1)=\frac{f(1)}{g(1)}=\frac{5}{2}\). The denominator value is (2), so division is allowed.