Evaluating Functions - Pre Algebra

1. Given $f(x) = 2x^2 - x + 3$, which expression represents $f(x+1) - f(x)$?
   Answer: $4x + 1$ When you encounter function composition problems like this, you're working with substitution and algebraic simplification. The key is to carefully substitute the new input into the original function, then perform the required operations. To find $f(x+1) - f(x)$, you first need to calculate $f(x+1)$ by substituting $(x+1)$ everywhere you see $x$ in the original function. Starting with $f(x) = 2x^2 - x + 3$: $f(x+1) = 2(x+1)^2 - (x+1) + 3$ Expanding $(x+1)^2 = x^2 + 2x + 1$: $f(x+1) = 2(x^2 + 2x + 1) - (x+1) + 3$ $f(x+1) = 2x^2 + 4x + 2 - x - 1 + 3$ $f(x+1) = 2x^2 + 3x + 4$ Now subtract $f(x)$: $f(x+1) - f(x) = (2x^2 + 3x + 4) - (2x^2 - x + 3)$ $= 2x^2 + 3x + 4 - 2x^2 + x - 3$ $= 4x + 1$ Choice A ($4x^2 + 3x + 2$) likely comes from forgetting to subtract $f(x)$ completely. Choice B ($4x + 3$) results from arithmetic errors in the constant terms. Choice C ($2x + 1$) suggests mistakes when combining like terms, possibly missing some $x$ terms during subtraction. Remember that difference problems like $f(x+1) - f(x)$ often simplify dramatically because many terms cancel out. Always distribute carefully when expanding, and double-check your arithmetic when combining like terms—the quadratic terms should disappear entirely in this type of problem.
2. The function $g(x) = x^2 - 4x + 1$ models the height of a ball in feet after $x$ seconds. At what time(s) other than $t = 1$ second is the ball at the same height as it was at $t = 1$ second?
   Answer: $t = 3$ seconds only First find $g(1) = 1 - 4 + 1 = -2$. Then solve $x^2 - 4x + 1 = -2$, which gives $x^2 - 4x + 3 = 0$. Factoring: $(x-1)(x-3) = 0$, so $x = 1$ or $x = 3$. Since we want times other than $t = 1$, the answer is $t = 3$. Choice A uses the axis of symmetry $x = 2$. Choice C solves $x^2 - 4x + 1 = 1$. Choice D includes the original time $t = 1$.
3. A piecewise function is defined as $f(x) = \begin{cases} x + 5 & \text{if } x \leq 2 \\ 3x - 1 & \text{if } x > 2 \end{cases}$. What is the value of $f(f(1))$?
   Answer: 17 First find $f(1)$. Since $1 \leq 2$, use the first piece: $f(1) = 1 + 5 = 6$. Then find $f(6)$. Since $6 > 2$, use the second piece: $f(6) = 3(6) - 1 = 17$. Choice A is just $f(1) - 1$. Choice B uses the wrong piece for $f(6)$: $6 + 5 - 3 = 8$. Choice D results from using $3(1) - 1 = 2$ then $2 + 5 + 4 = 11$.
4. A function $f$ is defined by $f(x) = 3x - 7$. If $f(a) = 8$, what is the value of $f(2a)$?
   Answer: 23 First, find $a$ by solving $f(a) = 8$: $3a - 7 = 8$, so $3a = 15$ and $a = 5$. Then $f(2a) = f(10) = 3(10) - 7 = 23$. Choice A results from calculating $2 \cdot 8 = 16$. Choice C comes from finding $a$ correctly but calculating $f(a+a)$ incorrectly. Choice D results from $f(2) \cdot f(a) = -1 \cdot 8$.
5. If Maria's bill last month was $63, how much would her bill be this month if she uses 40 fewer minutes than last month?
   Answer: $57.00 First, find last month's minutes: $45 + 0.15m = 63$, so $0.15m = 18$ and $m = 120$. This month she uses $120 - 40 = 80$ minutes beyond basic. So $B(80) = 45 + 0.15(80) = 45 + 12 = 57$. Choice B uses 95 minutes (adding instead of subtracting 40). Choice C uses 110 minutes (subtracting only 10). Choice D subtracts $6 directly from the original bill.

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