Function Notation & Evaluation - Accuplacer Aaf

The function $f(x) = ax^2 - 6x + 1$ is given, where $a$ is a constant. If $f(-2) = 25$ , what is the value of $a$ ?
Answer: 3 Substitute $x = -2$ and $f(-2) = 25$ into the function's definition and solve for $a$ : $25 = a(-2)^2 - 6(-2) + 1$ . Simplify the equation: $25 = a(4) + 12 + 1$ . This becomes $25 = 4a + 13$ . Subtract 13 from both sides: $12 = 4a$ . Divide by 4: $a = 3$ .
The function $f$ is defined by $f(x) = \begin{cases} 2x+5 & \text{if } x < 0 \\ x^2-1 & \text{if } x \geq 0 \end{cases}$ . What is the value of $f(-3) + f(3)$ ?
Answer: 7 To evaluate $f(-3)$ , we use the first rule since $-3 < 0$ : $f(-3) = 2(-3) + 5 = -6 + 5 = -1$ . To evaluate $f(3)$ , we use the second rule since $3 \geq 0$ : $f(3) = 3^2 - 1 = 9 - 1 = 8$ . The sum is $f(-3) + f(3) = -1 + 8 = 7$ .
Let $f(x) = (x-3)^2 + 4$ . For which of the following values of $x$ is $f(x) = 20$ ?
Answer: -1 Set $f(x)$ equal to 20 and solve for $x$ : $(x-3)^2 + 4 = 20$ . Subtract 4 from both sides: $(x-3)^2 = 16$ . Take the square root of both sides, remembering both positive and negative roots: $x-3 = 4$ or $x-3 = -4$ . Solve each equation: $x = 7$ or $x = -1$ . Of the choices provided, -1 is a correct value.
A function $k$ is defined by $k(n) = 3n - 2$ for all integers $n$ . If $k(m) + k(m + 1) = 23$ , what is the value of $m$ ?
Answer: $4$ First find expressions: $k(m) = 3m - 2$ and $k(m + 1) = 3(m + 1) - 2 = 3m + 3 - 2 = 3m + 1$ . Setting up the equation: $k(m) + k(m + 1) = (3m - 2) + (3m + 1) = 6m - 1 = 23$ . Solving: $6m = 24$ , so $m = 4$ . Verification: $k(4) = 10$ , $k(5) = 13$ , and $10 + 13 = 23$ ✓. Choice A gives $k(3) + k(4) = 7 + 10 = 17$ . Choice C gives $k(5) + k(6) = 13 + 16 = 29$ . Choice D gives $k(6) + k(7) = 16 + 19 = 35$ .
The function $f(x)$ is an odd function, meaning $f(-x) = -f(x)$ for all $x$ . If $f(2) = 5$ and $f(3)=10$ , what is the value of $f(-2) + f(3)$ ?
Answer: 5 We need to find $f(-2) + f(3)$ . We are given $f(3)=10$ . Since $f(x)$ is an odd function, we know that $f(-x) = -f(x)$ . Therefore, $f(-2) = -f(2)$ . We are given that $f(2)=5$ , so $f(-2) = -5$ . The sum is $f(-2) + f(3) = -5 + 10 = 5$ .
If $f(x) = x^2 - 10$ , what is the value of $f(f(3))$ ?
Answer: -9 This is a composition of a function with itself. First, evaluate the inner function: $f(3) = (3)^2 - 10 = 9 - 10 = -1$ . Then, use this result as the input for the outer function: $f(f(3)) = f(-1) = (-1)^2 - 10 = 1 - 10 = -9$ .
If $p(x) = x^2 - 4x + 7$ and $p(a) = 12$ , which of the following could be a value of $a$ ?
Answer: $5$ Set $p(a) = a^2 - 4a + 7 = 12$ , so $a^2 - 4a - 5 = 0$ . Factoring: $(a - 5)(a + 1) = 0$ , giving $a = 5$ or $a = -1$ . Checking the choices: Both $p(-1) = 1 + 4 + 7 = 12$ and $p(5) = 25 - 20 + 7 = 12$ are correct. Since both A and D are mathematically valid, either could be correct. Choice B gives $p(2) = 4 - 8 + 7 = 3$ . Choice C gives $p(3) = 9 - 12 + 7 = 4$ .
Given $g(x) = |2x - 6|$ , for which value of $x$ is $g(x) = g(x + 2)$ ?
Answer: $x = 2$ We need $|2x - 6| = |2(x + 2) - 6| = |2x + 4 - 6| = |2x - 2|$ . So $|2x - 6| = |2x - 2|$ . This equation holds when $2x - 6 = 2x - 2$ (impossible: $-6 = -2$ ) or when $2x - 6 = -(2x - 2) = -2x + 2$ . From the second case: $2x - 6 = -2x + 2$ , so $4x = 8$ and $x = 2$ . Verification: $g(2) = |4 - 6| = 2$ and $g(4) = |8 - 6| = 2$ ✓. Choice A gives $g(1) = 4$ and $g(3) = 0$ . Choice C gives $g(3) = 0$ and $g(5) = 4$ . Choice D gives $g(4) = 2$ and $g(6) = 6$ .
Let $f(x) = 4x - 7$ . The function $g$ is defined as $g(x) = f(x+2) - f(x)$ . What is the value of $g(10)$ ?
Answer: 8 To find $g(10)$ , we calculate $f(10+2) - f(10)$ , which is $f(12) - f(10)$ . First, calculate $f(12) = 4(12) - 7 = 48 - 7 = 41$ . Next, calculate $f(10) = 4(10) - 7 = 40 - 7 = 33$ . Finally, find the difference: $g(10) = 41 - 33 = 8$ . Alternatively, one could simplify $g(x)$ algebraically: $g(x) = (4(x+2)-7) - (4x-7) = (4x+8-7) - (4x-7) = 4x+1-4x+7 = 8$ . This shows that $g(x)=8$ for any value of $x$ .
Given $f(x) = \frac{2x + 1}{x - 3}$ , which expression represents $f(x + 2)$ ?
Answer: $\frac{2x + 5}{x - 1}$ To find $f(x + 2)$ , substitute $(x + 2)$ for every $x$ in the original function: $f(x + 2) = \frac{2(x + 2) + 1}{(x + 2) - 3} = \frac{2x + 4 + 1}{x + 2 - 3} = \frac{2x + 5}{x - 1}$ . Choice B incorrectly distributes the 2 in the numerator. Choice C forgets to add 4 when distributing 2(x + 2). Choice D correctly computes the numerator but incorrectly subtracts in the denominator.
If $f(x) = 1 - \frac{1}{x}$ , which of the following expressions is equivalent to $f(1-x)$ for $x \neq 1$ ?
Answer: $\frac{x}{x-1}$ To find $f(1-x)$ , substitute $(1-x)$ for $x$ in the expression for $f(x)$ : $f(1-x) = 1 - \frac{1}{1-x}$ . To simplify, find a common denominator: $f(1-x) = \frac{1-x}{1-x} - \frac{1}{1-x} = \frac{(1-x)-1}{1-x} = \frac{-x}{1-x}$ . Multiplying the numerator and denominator by -1 gives $\frac{x}{-(1-x)} = \frac{x}{x-1}$ .
The profit, $P$ , in dollars, from producing $x$ units of a product is given by the function $P(x) = 20x - (100 + 0.1x^2)$ . Which of the following best interprets the expression $P(50) - P(49)$ ?
Answer: The approximate profit gained by producing the 50th unit. The expression $P(50)$ represents the total profit from producing 50 units, and $P(49)$ is the total profit from producing 49 units. The difference, $P(50) - P(49)$ , represents the change in profit when production increases from 49 to 50 units. Therefore, this value is the additional, or marginal, profit gained from producing the 50th unit specifically.
Let $h(x) = \frac{4}{x-3}$ . If $h(a) = 4$ , what is the value of $h(a+1)$ ?
Answer: 2 This is a two-step problem. First, use the given information, $h(a) = 4$ , to find the value of $a$ . Substitute $a$ into the function: $\frac{4}{a-3} = 4$ . Multiply both sides by $a-3$ to get $4 = 4(a-3)$ . Divide by 4 to get $1 = a-3$ . Therefore, $a = 4$ . The question asks for $h(a+1)$ , which is $h(4+1) = h(5)$ . Now, evaluate the function at $x=5$ : $h(5) = \frac{4}{5-3} = \frac{4}{2} = 2$ .
Let $f(x) = 2x^2 + 3x - 1$ . If $f(a) = f(b)$ and $a \neq b$ , which expression represents the relationship between $a$ and $b$ ?
Answer: $a + b = -\frac{3}{2}$ If $f(a) = f(b)$ , then $2a^2 + 3a - 1 = 2b^2 + 3b - 1$ . Simplifying: $2a^2 + 3a = 2b^2 + 3b$ , so $2a^2 - 2b^2 + 3a - 3b = 0$ . Factoring: $2(a^2 - b^2) + 3(a - b) = 0$ , which gives $2(a - b)(a + b) + 3(a - b) = 0$ . Since $a \neq b$ , we have $a - b \neq 0$ , so we can divide by $(a - b)$ : $2(a + b) + 3 = 0$ , giving $a + b = -\frac{3}{2}$ . This is because the axis of symmetry of the parabola is $x = -\frac{3}{4}$ , and points equidistant from this axis have the same function value. Choice B has the wrong sign. Choices C and D represent different relationships.
A function $s$ satisfies $s(x + 3) = s(x) + 6$ for all real numbers $x$ . If $s(2) = 5$ , what is $s(11)$ ?
Answer: $23$ When you encounter a functional equation like $s(x + 3) = s(x) + 6$ , you're dealing with a pattern that describes how the function behaves as its input changes. This equation tells you that whenever you increase the input by 3, the output increases by 6. To find $s(11)$ starting from $s(2) = 5$ , you need to figure out how to get from an input of 2 to an input of 11. Since $11 - 2 = 9$ , and $9 = 3 \times 3$ , you need to apply the given relationship three times. Starting with $s(2) = 5$ : - $s(2 + 3) = s(5) = s(2) + 6 = 5 + 6 = 11$ - $s(5 + 3) = s(8) = s(5) + 6 = 11 + 6 = 17$ - $s(8 + 3) = s(11) = s(8) + 6 = 17 + 6 = 23$ Therefore, $s(11) = 23$ , which is answer C. Answer A (17) is the value of $s(8)$ , stopping one step too early. Answer B (29) might result from incorrectly adding 6 four times instead of three: $5 + 4 \times 6 = 29$ . Answer D (35) could come from miscounting and adding 6 five times: $5 + 5 \times 6 = 35$ . When working with functional equations, always count carefully how many steps you need to take. Calculate the difference between your starting and ending inputs, then divide by the step size to determine how many applications of the rule you need.
Given the functions $f(x) = x^2 + 5$ and $g(x) = 2x - 1$ , what is the value of $(f-g)(-3)$ ?
Answer: 21 The notation $(f-g)(-3)$ means $f(-3) - g(-3)$ . First, evaluate each function at $x=-3$ . $f(-3) = (-3)^2 + 5 = 9 + 5 = 14$ . $g(-3) = 2(-3) - 1 = -6 - 1 = -7$ . Finally, subtract the results: $14 - (-7) = 14 + 7 = 21$ .
The function $g(x)$ is an even function, meaning $g(-x) = g(x)$ for all $x$ . If $g(5) = -8$ , and $h(x) = 2g(x) + 3$ , what is the value of $h(-5)$ ?
Answer: -13 To find $h(-5)$ , we substitute $-5$ into the definition of $h(x)$ , which gives $h(-5) = 2g(-5) + 3$ . Since $g(x)$ is an even function, $g(-5) = g(5)$ . We are given that $g(5) = -8$ , so $g(-5) = -8$ . Substituting this value back gives $h(-5) = 2(-8) + 3 = -16 + 3 = -13$ .
For the function $w(t) = \frac{t}{t^2 + 1}$ , which statement about $w(-t)$ is correct?
Answer: $w(-t) = -w(t)$ for all values of $t$ When you encounter a function and need to evaluate what happens when you substitute $-t$ for $t$ , you're exploring function symmetry properties. This type of question tests whether you can recognize odd and even functions. To find $w(-t)$ , substitute $-t$ into the original function: $w(-t) = \frac{-t}{(-t)^2 + 1} = \frac{-t}{t^2 + 1}$ . Since $(-t)^2 = t^2$ , the denominator remains unchanged. Notice that $w(-t) = \frac{-t}{t^2 + 1} = -\frac{t}{t^2 + 1} = -w(t)$ . This means the function satisfies the definition of an odd function, where $w(-t) = -w(t)$ . Choice A claims $w(-t) = w(t)$ , which would make this an even function. However, we showed that $w(-t) = -w(t)$ , not $w(t)$ . Choice B suggests $w(-t) = w(t) + 2t$ . Substituting our results: $-w(t) = w(t) + 2t$ , which would mean $-2w(t) = 2t$ , or $w(t) = -t$ . This contradicts our original function. Choice C proposes $w(-t) = \frac{1}{w(t)}$ , meaning $-w(t) = \frac{1}{w(t)}$ , which would require $w(t) \cdot (-w(t)) = 1$ , or $-[w(t)]^2 = 1$ . Since squares are non-negative, this is impossible. Remember: when substituting $-t$ into rational functions, carefully track how the negative sign affects the numerator versus denominator. Even powers eliminate the negative sign, while odd powers preserve it.
If $f(x) = 2x-5$ and $g(x) = x^2$ , what is the value of $g(f(1)) - f(g(1))$ ?
Answer: 12 This problem requires evaluating two composite functions and finding their difference. First, find $g(f(1))$ . $f(1) = 2(1)-5 = -3$ . So, $g(f(1)) = g(-3) = (-3)^2 = 9$ . Next, find $f(g(1))$ . $g(1) = 1^2 = 1$ . So, $f(g(1)) = f(1) = 2(1)-5 = -3$ . Finally, calculate the difference: $g(f(1)) - f(g(1)) = 9 - (-3) = 9 + 3 = 12$ .
The function $f(x) = \frac{2x - 1}{3}$ is one-to-one. What is the value of $f^{-1}(5)$ ?
Answer: 8 Finding $f^{-1}(5)$ is equivalent to finding the value of $x$ such that $f(x) = 5$ . Set up the equation: $\frac{2x - 1}{3} = 5$ . Multiply both sides by 3 to get $2x - 1 = 15$ . Add 1 to both sides to get $2x = 16$ . Divide by 2 to get $x = 8$ .

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