Functions - GED Math

1. What is the zero of $f(x) = 2x - 8$?

   $x = 4$. A zero is the input that makes $f(x) = 0$: $2x - 8 = 0$.

2. What is the $x$-coordinate of the vertex of $f(x) = x^2 - 6x + 2$?

   $3$. Use $x = -\frac{b}{2a} = -\frac{-6}{2(1)} = 3$.

3. If $f(x) = x^2 - 2x$, what is $f(-3)$?

   $15$. $(-3)^2 - 2(-3) = 9 + 6 = 15$.

4. What does the vertical line test check on a graph?

   Whether the graph is a function: it is one only if no vertical line crosses it more than once.

5. What is the $y$-intercept of $f(x) = ax^2 + bx + c$?

   $c$, because $f(0) = a(0)^2 + b(0) + c = c$.

6. What makes a relation a function?

   Each input ($x$ value) has exactly one output ($y$ value).

7. How does a table of values show that a function is linear rather than quadratic?

   A linear function has constant first differences in its outputs (for equal steps in $x$); a quadratic has constant second differences.

8. In $f(x) = ax^2 + bx + c$, what does the sign of $a$ tell you?

   The direction the parabola opens: up (with a minimum) if $a > 0$, down (with a maximum) if $a < 0$.

9. If $f(x) = 4x - 1$, what is $f(3)$?

   $11$. Replace $x$ with $3$: $4(3) - 1 = 11$.

10. What are the domain and the range of a function?

    The domain is the set of all allowed inputs; the range is the set of all resulting outputs.

11. If $g(x) = 2x + 5$, for what value of $x$ does $g(x) = 17$?

    $x = 6$. Set $2x + 5 = 17$, so $2x = 12$.

12. In the linear function $f(x) = mx + b$, what do $m$ and $b$ represent?

    $m$ is the rate of change (slope) and $b$ is the starting value, the output when $x = 0$.

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