PreCalculus Final Exam - December 2024

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

For the relation given, does the first set of inputs and outputs define a function? Explain your reasoning.

No, it does not define a function because the input '3' is associated with multiple outputs.

What is the value of f(-a) for the function $f(x) = \sqrt{x - 2}$?

The value is $\sqrt{-a - 2}$.

Using the x-intercept method, identify one real solution of the equation: $x^3 + 10x^2 + 27x + 18 = 0$.

One solution is $x = -2$.

What are the solutions of the equation $x^2 - 4x - 5 = 0$?

The solutions are $x = 5$ and $x = -1$. Signup and view all the answers

Solve the equation $-3x^2 + 8x - 4 = 0$ for x.

The solutions are $x = 2$ and $x = \frac{2}{3}$. Signup and view all the answers

What is the solution to the equation $3(x + 1)^2 - 36 = 0$?

The solution is $x = 3$ or $x = -5$. Signup and view all the answers

What is the domain of the function $h(x) = \frac{4x}{x(x^2 - 81)}$?

The domain is $(-\infty, -9) \cup (-9, 0) \cup (0, 9) \cup (9, +\infty)$. Signup and view all the answers

Express the solution to the inequality $-10 \leq 2x - 4 \leq 0$ in interval notation.

The solution is $[-3, 5]$. Signup and view all the answers

Determine the x-intercepts of the quadratic function $f(x) = x^2 + 7x + 10$. What is the vertex of this graph?

The x-intercepts are $-2$ and $-5$, and the vertex is at $(-rac{7}{2}, -rac{9}{4})$. Signup and view all the answers

If the parent function is $f(x) = x^2$, what is the new rule for the function after translating it 4 units right and 3 units up?

The new function is $f(x) = (x - 4)^2 + 3$. Signup and view all the answers

Write the vertex form of the function $f(x) = -3x^2 + 12x - 7$. What is the vertex?

The vertex form is $f(x) = -3(x - 2)^2 + 5$, and the vertex is $(2, 5)$. Signup and view all the answers

Calculate $h(2+t)$ for the function $h(t) = t^3 + 3t$. What is the resulting polynomial?

The result is $t^3 + 6t^2 + 15t + 14$. Signup and view all the answers

For the function $f(x) = -2x$, what is the average rate of change between $x = 1$ and $x$?

The average rate of change is $-2$. Signup and view all the answers

Using the Remainder Theorem, what is the remainder when dividing $f(x)$ by $x - c$?

The remainder is $f(c)$, evaluated at the specific value of $c$. Signup and view all the answers

What is the maximum number of zeros a polynomial function can have and how can Descartes' Rule of Signs be applied?

The maximum number of zeros is $n$, where $n$ is the degree of the polynomial, with positive and negative zero estimates given by Descartes' Rule. Signup and view all the answers

List the potential rational zeros of a polynomial function without finding the actual zeros.

The potential rational zeros are factors of the constant term divided by factors of the leading coefficient. Signup and view all the answers

Flashcards

Function Relation

A relation is a function if each input value corresponds to exactly one output value.

Finding f(-a)

To find f(-a) , substitute -a for the variable x in the function's equation.

X-intercept Method

The x-intercept method involves setting the equation equal to zero and then solving for x. The solutions are the x-intercepts of the function's graph.