🧮 algebra

1. **State the problem:** We need to find the value of $\left(f+g\right)(3)$ using the graphs of the functions $f(x)$ and $g(x)$. 2. **Recall the formula:** The sum of two function

1. **Stating the problem:** We are given the function $$f(x) = -\frac{1}{2} x^3$$ and need to find its inverse function $$f^{-1}(x)$$. 2. **Recall the formula and rules:** To find

1. The problem asks if the given graph represents a function that has an inverse function. 2. A function has an inverse if and only if it is one-to-one (passes the Horizontal Line

1. **Simplify** $\sqrt{22m^7} \cdot \sqrt{8m^2}$. We use the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

1. The problem is to find the factorial of a number, which is the product of all positive integers up to that number. 2. The factorial of a number $n$ is denoted as $n!$ and is def

1. The problem asks us to classify given numbers as either rational or irrational. 2. A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\

1. **State the problem:** Find the value of $x$ that satisfies the equation $$3(x - 6) - 8x = -2 + 5(2x + 1).$$ 2. **Expand both sides:**

1. **State the problem:** Solve the equation $$\frac{3}{2}y + 25 = \frac{7}{3}y + 10$$ for $y$. 2. **Write down the equation:**

1. **State the problem:** Solve the equation $$9^{4x^2 - 1} = 1$$ for $x$. 2. **Recall the rule:** Any number (except 0) raised to the power 0 equals 1, i.e., $$a^0 = 1$$ for $a \n

1. **State the problem:** Fit a quadratic function in vertex form to the data points: $(-3,-5), (-2,-2), (0,1), (1,2), (2,1.5), (3,0), (3.5,-1), (4,-2)$, where the parabola opens d

1. The problem asks to solve the equation related to the jump distance: \( (4x - 9) = 2x + 2x \).\n2. This is an algebraic equation where we need to find the value of \( x \).\n3.

1. **Problem a:** Solve the equation $x(x - 5) = 6$. 2. **Formula and rules:** Expand the left side and bring all terms to one side to form a quadratic equation: $$x^2 - 5x = 6 \im

1. **State the problem:** Rewrite the expression $8 \cdot 8^6$ using a single exponent. 2. **Recall the exponent rule:** When multiplying powers with the same base, add the exponen

1. **State the problem:** Jessica used a total of $4 \frac{3}{4}$ gallons of gas. Each hour she drove, she used $\frac{5}{6}$ gallons. We need to find the total number of hours she

1. **State the problem:** We are given three inequalities involving $x$ and asked to understand their graph representations.

1. The problem states that the graph of $g(x)$ is a translation of $f(x) = |x|$. 2. The general form for a translated absolute value function is $$g(x) = a|x - h| + k$$ where:

1. The problem states that the graph of $g(x)$ is a translation of $f(x) = x^2$. 2. The general form for a translated parabola is $$g(x) = a(x - h)^2 + k$$ where $a$ controls the v

1. **State the problem:** Calculate the value of $\frac{4}{7} \div \frac{1}{2}$. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by

1. **State the problem:** Solve the quadratic equation by factoring: $$6x^2 - 5x + 6 = 6x + 3$$

1. **State the problem:** Solve the quadratic equation by factoring: $$2x^2 + 9x - 1 = 2x - 7$$ 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **State the problem:** We need to determine which of the given equations are true when $x=8$. 2. **List the equations:**