Functions and Function Notation - Quantitative Reasoning

Let $f(x)=x+4$ and $g(x)=\dfrac{x}{2}$ . What is the value of $f(g(10))$ ?
Answer: $9$ This question tests functions and function notation, specifically composition. The functions are f(x) = x + 4 and g(x) = x/2, so f(g(10)) means first apply g to 10, then f to that result. Compute g(10) = 10/2 = 5. Then f(5) = 5 + 4 = 9. Therefore, the value is 9. A tempting mistake might be to compose in reverse, g(f(10)) = (10+4)/2 = 7, option D. Another error could be adding before dividing, like (10 + 4)/2 = 7 again.
For all real numbers $x$ , a function $h$ is defined by $h(x)=\dfrac{2x+5}{x-1}$ , where $x\ne 1$ . What is the value of $h(3)$ ?
Answer: $\dfrac{11}{2}$ This question tests functions and function notation. The function h is defined by h(x) = (2x + 5)/(x - 1), where x ≠ 1, meaning you compute twice the input plus 5 over the input minus 1. To find h(3), replace x with 3 in the expression. So, numerator is 2*3 + 5 = 11, denominator is 3 - 1 = 2, so 11/2. Therefore, the value is 11/2. A tempting mistake might be to swap numerator and denominator, giving 2/11, which is option B. Another error could be miscalculating numerator as 2*3 +1=7, giving 7/2, option C.
A function $m$ is defined by $m(x)=|x-5|$ . What is the value of $m(2)$ ?
Answer: $3$ This question tests understanding of functions and function notation involving absolute value. The function is defined by m(x) = |x - 5|, which gives the absolute value of the difference between x and 5. To find m(2), we substitute x = 2: m(2) = |2 - 5| = |-3| = 3. Therefore, m(2) = 3. A common error would be to forget the absolute value and answer -3, but absolute value always returns a non-negative result, making the correct answer 3.
A function $f$ is defined by $f(x)=3x-5$ . What is the value of $f(4)$ ?
Answer: $7$ This question tests understanding of functions and function notation, specifically evaluating a linear function at a given input. The function is defined by f(x) = 3x - 5, which means to find f(x), we multiply the input by 3 and then subtract 5. To find f(4), we substitute x = 4 into the function: f(4) = 3(4) - 5 = 12 - 5 = 7. Therefore, f(4) = 7. A common error would be to confuse the order of operations or to substitute incorrectly, but following the function definition step by step yields the correct answer of 7.
Two functions are defined by $f(x)=x-5$ and $g(x)=\dfrac{1}{x}$ . What is the value of $g(f(2))$ ?
Answer: $-\dfrac{1}{3}$ This question tests understanding of composite functions and function notation. We have f(x) = x - 5 and g(x) = 1/x, and we need to find g(f(2)). First, we find f(2): f(2) = 2 - 5 = -3. Then we use this result as the input for g: g(-3) = 1/(-3) = -1/3. Therefore, g(f(2)) = -1/3. A common error would be to compute f(g(2)) instead, or to incorrectly handle the negative sign when taking the reciprocal of -3.
A function $f$ is defined by $f(x)=x^2-1$ . Which of the following equals $f(x+1)$ ?
Answer: $x^2+2x$ This question tests functions and function notation. The function f is defined by f(x) = x² - 1, so to find f(x+1), substitute x+1 for x. Replace x with x+1: (x+1)² - 1. Expand to x² + 2x + 1 - 1 = x² + 2x. Therefore, it equals x² + 2x. A tempting mistake might be to add 1 instead of subtracting, giving x² + 1, option A. Another error could be incomplete expansion, like x² + 2x - 1, option D.
Let $f(x)=x^2$ and $g(x)=2x-1$ . If $f(g(a))=g(f(a))$ , what is $a$ ?
Answer: 1 Substitute: f(g(a)) = $(2a - 1)^2$ and g(f(a)) = $2a^2$ - 1. Set them equal: $(2a - 1)^2$ = $2a^2$ - 1, which simplifies to 2 $(a - 1)^2$ = 0, so a = 1. The tempting wrong choice 0 fails because f(g(0)) = 1 while g(f(0)) = -1.
A function $g$ is defined by $g(x)=x^2-2x$ . Which of the following equals $g(-3)$ ?
Answer: $15$ This question tests understanding of functions and function notation, requiring evaluation of a quadratic function at a negative input. The function is defined by g(x) = x² - 2x, which means we square the input and subtract twice the input. To find g(-3), we substitute x = -3: g(-3) = (-3)² - 2(-3) = 9 - (-6) = 9 + 6 = 15. Therefore, g(-3) = 15. A tempting incorrect answer might be -3 (choice B), which could result from forgetting that (-3)² = 9 (not -9) or from sign errors when evaluating -2(-3).
A function $f$ is defined by $f(x)=3x-7$ . What is the value of $f(5)$ ?
Answer: $8$ This question tests functions and function notation. The function f is defined by f(x) = 3x - 7, which means you multiply the input by 3 and subtract 7 to get the output. To find f(5), replace x with 5 in the expression. So, compute 3 times 5 minus 7, which is 15 - 7. Therefore, the value is 8. A tempting mistake might be to forget to subtract 7, resulting in 15, which is option D. Another error could be misapplying the order, like subtracting first to get 5 - 7 = -2, which is option B.
A function $f$ is defined by $f(x)=x^2-6x+8$ . What is the value of $f(2)$ ?
Answer: $0$ This question tests functions and function notation. The function f is defined by f(x) = x² - 6x + 8, meaning square input, subtract 6 times input, add 8. To find f(2), replace x with 2 in the expression. So, 4 - 12 + 8 = -8 + 8 = 0. Therefore, the value is 0. A tempting mistake might be to miscalculate square as 2, giving 2-12+8=-2, not an option. Another error could be forgetting signs, like 4 +12 +8=24, not; option C -4 from 4-6*2=4-12=-8, forgetting +8.
A function $m$ is defined by $m(x)=|x-6|$ . What is the value of $m(2)$ ?
Answer: $4$ This question tests understanding of functions and function notation with absolute value. The function m is defined by m(x) = |x - 6|, which gives the absolute value of the difference between x and 6. To find m(2), we substitute 2 for x: m(2) = |2 - 6| = |-4| = 4. Therefore, m(2) = 4. A common mistake would be to forget to apply the absolute value and leave the answer as -4, or to incorrectly compute the subtraction inside the absolute value bars.
A function $f$ is defined as follows: for any number $x$ , $f(x)$ is the result of taking $x$ , multiplying by 2, and then adding 5. Which of the following equals $f(a-3)$ ?
Answer: $2a-1$ This question tests understanding of functions and function notation with a verbal description. The function f is defined as: f(x) = 2x + 5 (multiply input by 2, then add 5). To find f(a-3), we substitute (a-3) for x: f(a-3) = 2(a-3) + 5 = 2a - 6 + 5 = 2a - 1. Therefore, f(a-3) = 2a - 1. A common mistake would be to incorrectly distribute the 2 or to mishandle the algebraic simplification.
A function $q$ is defined as follows: $q(n)$ equals the square of $n$ minus 3. What is the value of $q(\tfrac{1}{2})$ ?
Answer: $-\tfrac{11}{4}$ This question tests functions and function notation. The function q is defined by q(n) = n² - 3, meaning square the input and subtract 3. To find q(1/2), replace n with 1/2 in the expression. So, (1/2)² = 1/4, then 1/4 - 3 = 1/4 - 12/4 = -11/4. Therefore, the value is -11/4. A tempting mistake might be to subtract first, like (1/2 - 3)² = (-5/2)² = 25/4, not an option. Another error could be using 1/2 squared as 1/2, giving 1/2 -3 = -5/2, option C.
A function $r$ is defined by $r(x)=|x-3|$ . What is the value of $r(-1)$ ?
Answer: $4$ This question tests functions and function notation. The function r is defined by r(x) = |x - 3|, meaning the absolute value of the input minus 3. To find r(-1), replace x with -1 in the expression. So, -1 - 3 = -4, and | -4 | = 4. Therefore, the value is 4. A tempting mistake might be to forget the absolute value, giving -4, option A. Another error could be subtracting in reverse, |3 - (-1)| = |4| = 4, same, but option D -2 might come from partial miscalculation.
For $f(x)=x^2-3x$ , which equals $\frac{f(x+h)-f(x)}{h}$ for $h \neq 0$ ?
Answer: $2x+h-3$ Expand f(x+h): $x^2$ + 2xh + $h^2$ - 3x - 3h. Subtract f(x)= $x^2$ -3x to get 2xh+ $h^2$ -3h. Factor out h and divide by h, with h not 0, giving 2x+h-3. The tempting error 2x+h+3 comes from treating -3(x+h) as -3x+3h; the -3h must be subtracted.
A function $f$ is defined by $f(x)=3x-7$ . What is the value of $f(4)$ ?
Answer: $5$ This question tests understanding of functions and function notation. The function f is defined by the rule f(x) = 3x - 7, which means to find f of any input, we multiply that input by 3 and then subtract 7. To find f(4), we substitute 4 for x in the expression: f(4) = 3(4) - 7 = 12 - 7 = 5. Therefore, f(4) = 5. A common error would be to confuse the input with the output or to misapply the arithmetic operations in the function rule.
If $f(x+1)=x^2-2x+3$ , what is $f(x)$ ?
Answer: $x^2-4x+6$ Let u = x + 1, so x = u - 1. Substituting gives f(u) = $(u - 1)^2$ - 2(u - 1) + 3 = $u^2$ - 4u + 6, so f(x) = $x^2$ - 4x + 6. The tempting $x^2$ - 2x + 3 is just the original expression f(x + 1), not f(x).
Functions $f$ and $g$ are defined by $f(x)=x-1$ and $g(x)=x^2$ . What is the value of $(g\circ f)(3)$ , where $(g\circ f)(x)=g(f(x))$ ?
Answer: $4$ This question tests understanding of function composition and function notation. The composition (g∘f)(x) means g(f(x)), where we first apply f, then apply g to the result. Given f(x) = x - 1 and g(x) = x², we first find f(3): f(3) = 3 - 1 = 2. Then we apply g to this result: g(f(3)) = g(2) = 2² = 4. Therefore, (g∘f)(3) = 4. A common error would be to compute f(g(3)) instead, which would give f(9) = 8.
Let $f(x)=2x+3$ and $g(x)=x^2$ . For $x=2$ , what is the value of $f(x)-g(x)$ ?
Answer: $3$ This question tests functions and function notation. The functions are f(x) = 2x + 3 and g(x) = x², so f(x) - g(x) means subtract g from f at x=2. Compute f(2) = 4 + 3 = 7, g(2) = 4. Then 7 - 4 = 3. Therefore, the value is 3. A tempting mistake might be to subtract in reverse, 4 - 7 = -3, option B. Another error could be adding them, 7 + 4 = 11, not an option.
A function $g$ is defined by $g(x)=x^2-2x$ . What is the value of $g(-3)$ ?
Answer: $15$ This question tests understanding of functions and function notation with a quadratic expression. The function g is defined by g(x) = x² - 2x, which means we square the input and subtract twice the input. To find g(-3), we substitute -3 for x: g(-3) = (-3)² - 2(-3) = 9 - (-6) = 9 + 6 = 15. Therefore, g(-3) = 15. A common mistake would be to incorrectly handle the negative signs, such as computing (-3)² as -9 instead of 9, or forgetting that -2(-3) = +6.

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