MAT 1580 Precalculus Test 1 Review

Flashcards covering key practice problems and concepts for MAT 1580 Precalculus Test 1, including function transformations, polynomial functions, asymptotes, quadratic functions, and inequalities.

What are the $x$- and $y$-intercepts of the graph of $y = \sqrt{x - 7}$?

The $x$-intercept is $(7, 0)$ and there is no $y$-intercept.

What are the $x$- and $y$-intercepts of the graph of $y = \sqrt{x + 4}$?

The $x$-intercept is $(-4, 0)$ and the $y$-intercept is $(0, 2)$.

What symmetry and intercepts does the graph of $y = -\sqrt{25 - x^2}$ possess?

The graph is symmetric with respect to the $y$-axis. Its $x$-intercepts are $(-5, 0)$ and $(5, 0)$, and its $y$-intercept is $(0, -5)$.

For the piecewise function $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \ge 0 \end{cases}$, what are the values of $f(-2)$, $f(-1)$, $f(0)$, $f(1)$, and $f(5)$?

$f(-2) = 4$, $f(-1) = 1$, $f(0) = 1$, $f(1) = 3$, and $f(5) = 11$.

For the function $f(x) = x^2 - 1$, what is the simplified difference quotient $\frac{f(a+h) - f(a)}{h}$ for $h \neq 0$?

$\frac{f(a+h) - f(a)}{h} = 2a + h$.

What is the domain of the function $f(x) = \frac{2x + 1}{x^2 + 1}$?

The domain is all real numbers, $(-\infty, \infty)$, because no value of $x$ makes the denominator equal to $0$.

What is the domain of the function $h(x) = \sqrt{2x^2 + 3x + 1}$?

The domain is $(-\infty, -1] \cup \left[-\frac{1}{2}, \infty\right)$.

Does the equation $x^2 + y^2 - 25 = 0$ define $y$ as a function of $x$?

No, because solving for $y$ gives $y = \pm\sqrt{25 - x^2}$, which assigns two values of $y$ for a given value of $x$.

For the function $h$ shown in the graph, what are the values of $h(-3)$, $h(-2)$, $h(2)$, and $h(4)$?

$h(-3) = 3$, $h(-2) = 1$, $h(2) = 3$, and $h(4) = 3$.

For the graph shown, on which intervals is the function increasing and decreasing?

Increasing on $[-1, 1]$ and $[3, \infty)$; decreasing on $(-\infty, -1]$ and $[1, 3]$.

What piecewise-defined function corresponds to the graph shown?

$$f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ 2 & \text{if } 0 \le x \le 1 \\ -x + 6 & \text{if } x > 1 \end{cases}$$

What transformations of the standard function $y = |x|$ produce the graph shown?

Shift one unit to the right, reflect about the $x$-axis, then shift upward $2$ units.

Is the function $f(x) = \frac{x^4 - 2}{x^5 - 5x}$ even, odd, or neither?

It is odd because $f(-x) = -f(x)$, meaning its graph is symmetric with respect to the origin.

Given $f(x) = \sqrt{2x - 1}$ and $g(x) = \sqrt{3x + 5}$, what are $(f+g)(x)$ and $(f/g)(x)$ and their respective domains?

$(f+g)(x) = \sqrt{2x - 1} + \sqrt{3x + 5}$ with domain $\left[\frac{1}{2}, \infty\right)$; $(f/g)(x) = \frac{\sqrt{2x - 1}}{\sqrt{3x + 5}}$ with domain $\left[\frac{1}{2}, \infty\right)$.

Given $f(x) = \frac{1}{x+2}$ and $g(x) = \frac{1}{x-2}$, what is $(f \circ g)(x)$ and its domain?

$(f \circ g)(x) = \frac{x-2}{2x-3}$ with domain $\left\{x \mid x \neq 2, x \neq \frac{3}{2}\right\}$.

Is the function $g(x) = x^2 - 1$ for $x \ge 0$ one-to-one?

Yes, because for any $x_1, x_2 \ge 0$, if $g(x_1) = g(x_2)$, then $x_1^2 - 1 = x_2^2 - 1 \implies x_1 = x_2$.

What is the inverse function $f^{-1}(x)$ of $f(x) = \frac{1}{2 - x}$ for $x \neq 2$?

$f^{-1}(x) = 2 - \frac{1}{x} = \frac{2x - 1}{x}$ for $x \neq 0$.

What is the vertex, axis of symmetry, and standard form equation of the quadratic function $f(x) = x^2 + 4x + 5$ shown in the graph?

Standard form: $f(x) = (x + 2)^2 + 1$; Vertex: $(-2, 1)$; Axis of symmetry: $x = -2$.

What is the vertex, axis of symmetry, and standard form equation of the quadratic function $f(x) = -3x^2 + 12x - 11$ shown in the graph?

Standard form: $f(x) = -3(x - 2)^2 + 1$; Vertex: $(2, 1)$; Axis of symmetry: $x = 2$.

What is the maximum or minimum value of the quadratic function $f(x) = 3x^2 + 9x + 14$?

The function has a minimum value of $\frac{29}{4}$ at $x = -\frac{3}{2}$.

What is the maximum or minimum value of the quadratic function $f(x) = -3x^2 + 12x - 3$?

The function has a maximum value of $9$ at $x = 2$.

If a ball is thrown directly upward with a velocity of $48\,\text{ft/s}$, its height in feet after $t$ seconds is given by $y = 48t - 16t^2$. What is the maximum height attained by the ball?

The maximum height attained by the ball is $36\,\text{ft}$.

For the revenue function $R(x) = 80x - 0.2x^2$, what is the maximum revenue and how many units $x$ should be manufactured to obtain it?

Manufacturing $200$ units yields the maximum revenue of $\$8\,000$.

What are the quotient and remainder when $2x^4 - x^3 + 16x^2 - 6$ is divided by $2x + 1$ using long division?

Quotient: $x^3 - x^2 + 8x - 4$; Remainder: $-2$.

Using synthetic division and the Remainder Theorem, what is $p(3)$ for $p(x) = 2x^3 - 2x^2 - 11x - 100$?

$p(3) = -31$.

How does the Factor Theorem demonstrate that $(x - 4)$ is a factor of $P(x) = x^5 + 4x^4 - 36x^3 - 16x^2 + 320x$?

Since $P(4) = 4^5 + 4(4^4) - 36(4^3) - 16(4^2) + 320(4) = 0$, $(x - 4)$ is a factor of $P(x)$.

What degree $4$ polynomial has zeros $-1$, $0$, $1$, and $5$?

$P(x) = x(x + 1)(x - 1)(x - 5) = x^4 - 5x^3 - x^2 + 5x$.

What are all the rational zeros of the polynomial $p(x) = x^3 - 6x^2 + 11x - 6$?

The rational zeros are $x = 1$, $x = 2$, and $x = 3$.

What is the complete factorization and zeros of $p(x) = x^6 - 6x^4 + 9x^2$?

Complete factorization: $p(x) = x^2(x - \sqrt{3})^2(x + \sqrt{3})^2$. The zeros are $x = 0$, $x = \sqrt{3}$, and $x = -\sqrt{3}$, each of multiplicity $2$.

What polynomial $P(x)$ of degree $3$ with integer coefficients has zeros $3$ and $1 - 3i$?

$P(x) = x^3 - 5x^2 + 16x - 30$.

What are all horizontal and vertical asymptotes of $r(x) = \frac{5}{x - 5}$?

Horizontal asymptote: $y = 0$; Vertical asymptote: $x = 5$.

What are all horizontal and vertical asymptotes of $r(x) = \frac{x^2 - 2x}{x^2 - 11x + 30}$?

Horizontal asymptote: $y = 1$; Vertical asymptotes: $x = 5$ and $x = 6$.

What is the solution set for the linear inequality $3 \le 4 - x < 14$?

In interval notation, the solution set is $[7, 18)$.

What is the solution set for the nonlinear inequality $x^2 + x - 20 > 0$?

In interval notation, the solution set is $(-\infty, -5) \cup (4, \infty)$.