Functions and Function Notation
Help Questions
GRE Quantitative Reasoning › Functions and Function Notation
Let $f(x)=2x+3$ and $g(x)=x^2$. For $x=2$, what is the value of $f(x)-g(x)$?
$-1$
$3$
$7$
$-3$
$1$
Explanation
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.
Two functions are defined by $p(x)=4-x$ and $q(x)=2x+1$. What is the value of $p(q(3))$?
$7$
$-9$
$-7$
$-3$
$3$
Explanation
This question tests understanding of composite functions and function notation. We have p(x) = 4 - x and q(x) = 2x + 1, and we need to find p(q(3)). First, we find q(3): q(3) = 2(3) + 1 = 6 + 1 = 7. Then we use this result as the input for p: p(7) = 4 - 7 = -3. Therefore, p(q(3)) = -3. A common error would be to compute q(p(3)) instead, which would give a different result, or to make arithmetic errors in the two-step process.
A function $g$ is defined by $g(x)=x^2+2x$. Which of the following equals $g(3y)$?
$9y^2+2y$
$9y^2+6y$
$3y^2+6y$
$(3y)^2+2y$
$3y^2+2y$
Explanation
This question tests functions and function notation. The function g is defined by g(x) = x² + 2x, so to find g(3y), substitute 3y for x. Replace x with 3y: (3y)² + 2*(3y). Compute 9y² + 6y. Therefore, it equals 9y² + 6y. A tempting mistake might be to add coefficients incorrectly, like 3y² + 2y, option B. Another error could be forgetting to square the 3, giving 3y² + 6y, option E.
A function $r$ is defined by $r(x)=x^2+1$. Which of the following equals $r(2k)$?
$k^2+1$
$(2k)^2$
$4k^2+1$
$2k^2+1$
$4k+1$
Explanation
This question tests understanding of functions and function notation with algebraic expressions. The function r is defined by r(x) = x² + 1. To find r(2k), we substitute 2k for x in the expression: r(2k) = (2k)² + 1 = 4k² + 1. Therefore, r(2k) = 4k² + 1. A common error would be to incorrectly square 2k as 2k² instead of 4k², forgetting that (2k)² = 2² × k² = 4k².
A function $q$ is defined as follows: $q(n)$ equals the square of $n$ minus 3. What is the value of $q(\tfrac{1}{2})$?
$-\tfrac{11}{4}$
$\tfrac{11}{4}$
$-\tfrac{5}{2}$
$\tfrac{5}{4}$
$-\tfrac{7}{4}$
Explanation
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 $t$ is defined by $t(x)=x^2-4$. What is the value of $t(a+1)$ in terms of $a$?
$a^2+2a-3$
$a^2-3$
$a^2+1-4$
$(a+1)^2-4$
$a^2+2a+5$
Explanation
This question tests understanding of functions and function notation with variable substitution. The function is defined by t(x) = x² - 4. To find t(a + 1), we substitute (a + 1) for x in the function: t(a + 1) = (a + 1)² - 4. Expanding (a + 1)²: (a + 1)² = a² + 2a + 1. Therefore, t(a + 1) = a² + 2a + 1 - 4 = a² + 2a - 3. A common error would be to expand (a + 1)² incorrectly as a² + 1, forgetting the middle term 2a from the binomial expansion.
A function $f$ is defined by $f(x)=3x-5$. What is the value of $f(4)$?
$-7$
$-17$
$3$
$17$
$7$
Explanation
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.
Functions $f$ and $g$ are defined by $f(x)=x^2+1$ and $g(x)=2x+3$. For $x=2$, what is the value of $f(x)-g(x)$?
$-2$
$4$
$0$
$2$
$-4$
Explanation
This question tests understanding of functions and function notation, specifically evaluating and subtracting two functions at a given point. We have f(x) = x² + 1 and g(x) = 2x + 3, and need to find f(2) - g(2). First, f(2) = 2² + 1 = 4 + 1 = 5. Next, g(2) = 2(2) + 3 = 4 + 3 = 7. Therefore, f(2) - g(2) = 5 - 7 = -2. A common error might be to subtract the functions first and then evaluate, but we need to evaluate each function separately at x = 2 before subtracting.
A function $h$ is defined by $h(t)=\dfrac{t-1}{t+3}$ for $t\ne -3$. What is the value of $h(1)$?
$\dfrac{2}{3}$
$\dfrac{2}{4}$
$\dfrac{1}{2}$
$\dfrac{1}{4}$
$0$
Explanation
This question tests understanding of functions and function notation with a rational expression. The function h is defined by h(t) = (t-1)/(t+3) for t ≠ -3. To find h(1), we substitute 1 for t in the expression: h(1) = (1-1)/(1+3) = 0/4 = 0. Therefore, h(1) = 0. A common error would be to simplify the fraction incorrectly or to confuse the numerator and denominator operations.
A function $r$ is defined by $r(x)=|x-3|$. What is the value of $r(-1)$?
$-4$
$3$
$2$
$-2$
$4$
Explanation
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.