Function Rules - Pre Algebra

1. A function has the rule $g(x) = 3x - 5$. If $g(a) = g(2a - 4)$, what is the value of $a$?
   Answer: $a = 4$ Since $g(a) = g(2a - 4)$, we have $3a - 5 = 3(2a - 4) - 5$. Simplifying the right side: $3a - 5 = 6a - 12 - 5 = 6a - 17$. Setting them equal: $3a - 5 = 6a - 17$. Solving: $-5 + 17 = 6a - 3a$, so $12 = 3a$, therefore $a = 4$. Choice A would give $g(2) = 1$ and $g(0) = -5$, which are not equal. Choice C would give $g(1) = -2$ and $g(-2) = -11$, which are not equal. Choice D would give $g(3) = 4$ and $g(2) = 1$, which are not equal.
2. A function $f(x)$ has the property that $f(x + 2) = f(x) + 6$ for all values of $x$. If $f(1) = 4$, what is $f(7)$?
   Answer: 22 This question tests your understanding of function properties and how to use given relationships to find unknown values. When you see a functional equation like $f(x + 2) = f(x) + 6$, you're being told that every time the input increases by 2, the output increases by 6. Starting with $f(1) = 4$, you can use the given property repeatedly to find $f(7)$. Since $f(x + 2) = f(x) + 6$, you have: - $f(3) = f(1 + 2) = f(1) + 6 = 4 + 6 = 10$ - $f(5) = f(3 + 2) = f(3) + 6 = 10 + 6 = 16$ - $f(7) = f(5 + 2) = f(5) + 6 = 16 + 6 = 22$ Therefore, $f(7) = 22$, which is choice D. Looking at the wrong answers: Choice A (18) results from adding 6 only twice instead of three times—you might get this by miscounting the steps from $x = 1$ to $x = 7$. Choice B (20) could come from incorrectly thinking you add 6 for each unit increase rather than each increase of 2. Choice C (16) stops one step short, giving you $f(5)$ instead of $f(7)$. When working with functional equations, always track your steps carefully and count how many times you need to apply the given property. The key insight is recognizing that going from $x = 1$ to $x = 7$ requires three applications of the rule since $7 = 1 + 3(2)$.
3. A function is described as: "If the input is positive, output the input minus 2. If the input is negative or zero, output the opposite of the input." Which piecewise function represents this rule?
   Answer: $f(x) = \begin{cases} x - 2 & \text{if } x > 0 \\ -x & \text{if } x \leq 0 \end{cases}$ When you encounter a piecewise function problem, you need to carefully translate each condition from words into mathematical notation, paying close attention to which inequality symbols to use. Let's break down the given rule: "If the input is positive, output the input minus 2. If the input is negative or zero, output the opposite of the input." The first condition says "if the input is positive," which means $x > 0$. For these inputs, we output $x - 2$. The second condition says "if the input is negative or zero," which means $x \leq 0$. For these inputs, we output "the opposite of the input," which is $-x$. So our function should be: $f(x) = \begin{cases} x - 2 & \text{if } x > 0 \\ -x & \text{if } x \leq 0 \end{cases}$ This matches option C exactly. Let's see why the other options fail: Option A uses $x \geq 0$ for the first condition, which would include zero. But the problem states zero should follow the second rule (output the opposite), not the first rule. Option B has the wrong operation in the first case: $x + 2$ instead of $x - 2$. This completely changes what the function does for positive inputs. Option D swaps the rules entirely—it applies the "opposite" rule to positive numbers and the "minus 2" rule to negative numbers and zero. Study tip: Always check where zero belongs when translating "positive" and "negative or zero" conditions. The word "positive" excludes zero, so use $>$, while "negative or zero" includes zero, so use $\leq$.
4. Consider the function rule $h(x) = |2x + 1| - 3$. In words, this rule can be described as:
   Answer: Take twice the input plus one, find its absolute value, then subtract three from the result The function $h(x) = |2x + 1| - 3$ follows this order of operations: first calculate $2x + 1$ (twice the input plus one), then take the absolute value of that result, then subtract 3. Choice B describes $2|x| + 1 - 3$. Choice C describes $|2(x + 1)| - 3 = |2x + 2| - 3$. Choice D describes $|2x + 1 - 3| = |2x - 2|$.
5. Based on the delivery company's pricing rule described above, what is the cost to ship a package weighing 6.5 pounds?
   Answer: $19.25 Since the package weighs 6.5 pounds, which is more than 2 pounds, we use the rule: $8 plus $2.50 for each pound over 2 pounds. The package is 6.5 - 2 = 4.5 pounds over the 2-pound threshold. The cost is $8 + $2.50 × 4.5 = $8 + $11.25 = $19.25.
6. Two students write function rules for the same relationship. Student A writes: "Triple the input and subtract 4." Student B writes: $g(x) = 3x - 4$. If both rules produce the same output when the input is $k$, but different outputs when the input is $k + 1$, what can be concluded?
   Answer: The premise is impossible since both descriptions represent the same mathematical function Student A's rule "Triple the input and subtract 4" translates exactly to $f(x) = 3x - 4$, which is identical to Student B's rule $g(x) = 3x - 4$. Since both functions are mathematically identical, they must produce the same output for any given input, including both $k$ and $k + 1$. The premise that they produce different outputs for $k + 1$ is mathematically impossible. Choices B and C suggest incorrect interpretations, but "triple the input and subtract 4" unambiguously means $3x - 4$. Choice A suggests an error but doesn't recognize that the functions are actually identical.
7. Maria writes a function rule: "The output is 4 more than twice the square of the input." Which equation correctly represents this function?
   Answer: $f(x) = 2x^2 + 4$ The phrase "twice the square of the input" means $2 \cdot x^2 = 2x^2$, and "4 more than" means adding 4, giving $f(x) = 2x^2 + 4$. Choice B represents "4 more than the square of twice the input" which would be $(2x)^2 + 4 = 4x^2 + 4$. Choice C represents "twice the quantity of the square of the input plus 4" which gives $2(x^2 + 4) = 2x^2 + 8$. Choice D reverses the coefficients.
8. A function rule states: "Subtract 3 from the input, then multiply the result by itself." If the output of this function is 16, what are the possible input values?
   Answer: $x = 7$ and $x = -1$ The function rule translates to $f(x) = (x - 3)^2$. Setting this equal to 16: $(x - 3)^2 = 16$. Taking the square root of both sides: $x - 3 = \pm 4$. This gives $x - 3 = 4$ or $x - 3 = -4$, so $x = 7$ or $x = -1$. Choice A only considers the positive square root. Choice B only considers the negative square root. Choice D incorrectly solves $x^2 = 16$ instead of $(x-3)^2 = 16$.

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