Algebra Competency Assessment

Simplify the following expression: $$\left(2x^2-3x+4\right)-\left(x^2+2x-5\right)$$ .

Include a step by step explanation of how you arrived at your solution.

Select all the expressions that are equivalent to $$x^2-6x+9$$

Solve for x. $$2x\ -\ \frac{3}{x\ }=5$$

Solve the inequality and choose all possible solutions.

$$\frac{2x-1}{x+3}>0$$

Which of the following is not a function? Why?

A. $$\left\{\left(2,1\right),\left(3,1\right),\left(4,1\right)\right\}$$

B. $$\left\{\left(1.2\right),\left(2,3\right),\ \left(3.4\right),\ \left(1,\ 5\right)\right\}$$

C. $$y=2x+1$$

D. $$x^3+4$$

Given $$f\left(x\right)\ =\ 3x-2$$ , find the inverse function.

Explain your steps for solving for x in terms of y, and then write $$f^{-1}\left(x\right)$$ .

Solve the logarithmic equation:

$$\log_2\left(x\right)\ +\ \log_2\left(x-2\right)\ =\ 3$$

Solve: $$3\left(x-4\right)\ =\ 2x+5$$

Cynthia's Solution:

3x - 4 = 2x + 5

3x - 2x = 5 + 4

x = 9

What mistake did Cynthia make when solving the equation?

"A plant grows 3 cm per day." A student models it as : $$h\left(t\right)\ =\ 3^t$$

What misunderstanding does the student demonstrate? How would you guide them to correct the model?

Given: $$f\left(x\right)\ =\ x^2$$ , student writes the transformation $$g\left(x\right)\ =\ \left(x-2\right)^2+3$$ as a reflection over the y-axis, then translated 2 units left and 3 up.

What errors are present in the student’s interpretation of the transformation? Provide the corrected interpretation.