🧮 algebra

1. **State the problem:** We need to show that the function $f(x) = x^4 + x - 1$ has a real root $\alpha$ in the interval $[0.5, 1.0]$. 2. **Evaluate $f(x)$ at the endpoints:**

1. **State the problem:** We want to find the value of $a$ such that the piecewise function $$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ ax + 1 & \text{if } x \geq 1 \end{cases

1. The problem asks us to identify which function is odd from the given options. 2. Recall that a function $f(x)$ is odd if it satisfies the condition:

1. The problem asks us to determine which of the given functions is even. 2. Recall that a function $f(x)$ is even if it satisfies the condition:

1. The problem asks to find the composition of functions $(f \circ g)(x)$, which means $f(g(x))$. 2. Given $f(x) = 2x + 1$ and $g(x) = x^2$, substitute $g(x)$ into $f$:

1. Problem: Simplify $$\sqrt{\frac{a-b}{a+b}} \sqrt{\frac{a^2+2ab+b^2}{a^2-b^2}}$$ where $$a>b>0$$. Step 1: Recognize $$a^2+2ab+b^2 = (a+b)^2$$ and $$a^2-b^2 = (a-b)(a+b)$$.

1. The problem asks for the range of the function $f(x) = x^2$. 2. The function $f(x) = x^2$ is a parabola opening upwards with vertex at $(0,0)$.

1. The problem is to determine the domain of the function $f(x) = \frac{1}{x - 3}$.\n\n2. The domain of a function includes all real numbers $x$ for which the function is defined.

1. The problem is to simplify or understand the expression $$M_2 (v_1^+ v_1 - v_2)$$. 2. First, clarify the notation: $v_1^+$ likely means the positive part or a specific operation

1. **State the problem:** Solve the system of equations: $$x + y = 17$$

1. The problem is to understand the equation $X + y = 17$ and explore its implications. 2. This is a linear equation in two variables, $X$ and $y$.

1. **State the problem:** Solve the equation $$\frac{2}{x+3} + \frac{5}{3-x} = \frac{6}{x^2 - 9}$$. 2. **Recognize the denominator:** Note that $$x^2 - 9 = (x+3)(x-3)$$.

1. The problem involves understanding summation notation, averages, linear relationships, and temperature conversions. 2. Summation properties:

1. **State the problem:** Simplify and solve the equation $$\frac{1}{2x+2} + \frac{5}{x^2-1} = \frac{1}{x-1}$$. 2. **Factor denominators:**

1. **State the problem:** A group of friends plans to rent a beach house for 9000. They want to split the cost equally, but one friend suggests each person pays the square root of

1. **State the problem:** A group of friends plans to rent a beach house for 9000. They want to split the cost equally, but one friend suggests each person's share should be $\sqrt

1. The problem is to solve the equation $$2x + 3 = 11$$ for $x$. 2. Start by isolating the variable term $2x$ on one side. Subtract 3 from both sides:

1. The problem is to solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. We start by factoring the quadratic expression. We look for two numbers that multiply to 6 and add to -5.

1. The problem is to solve the equation $$2x + 3 = 7$$ for $x$. 2. Start by isolating the variable term on one side. Subtract 3 from both sides:

1. The problem is to solve the equation $$2x + 3 = 11$$ for $x$. 2. Start by isolating the variable term $2x$ on one side. Subtract 3 from both sides:

1. **State the problem:** A group of friends is renting a beach house for 9000. They want to split the cost equally, but one friend suggests that each person's share should be equa