ACT Math Domain 1: Preparing for Higher Math

If the function \(f(x) = \frac{3x - 7}{2x + 5}\) is defined for all real numbers except \(x = -\frac{5}{2}\), what is \(f\left(\frac{1}{f(-3)}\right)\)?

If \(g(x) = x^3 - 4x^2 + x + 6\) and \(g(k) = 0\), which of the following could be a possible value of \(k\)?

The equation \(\frac{1}{x-1} + \frac{1}{x+1} = \frac{2}{x}\) is valid for all \(x\) except:

The matrix equation \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 11 \end{pmatrix}\) solves for:

What is the minimum value of the function \(h(t) = t^4 - 4t^3 + 6t^2 - 4t + 5\)?

A cube has a volume of \(8x^3\) cubic units. If each side of the cube is multiplied by 3, what is the new volume?

The function \(f(x) = a^x\) passes through the point (2,9). What is the value of \(a\)?

If the inequality \(|x - 5| > 7\) is true, which of the following represents all possible values of \(x\)?

What are the roots of the equation \(x^3 - 2x^2 - 5x + 6 = 0\)?

Given the sequence defined by the recursive formula \(a_{n+1} = 3a_n + 1\) with \(a_1 = 1\), what is \(a_4\)?

If the set \(S\) contains all solutions \(x\) to the inequality \(x^2 - 3x - 18 \leq 0\), what are the interval bounds of \(S\)?

If \(z\) and \(w\) are complex numbers such that \(z = 3 + 4i\) and \(w = 1 - 2i\), what is \(z \cdot w\)?

The sides of a triangle are given by \(x\), \(x+1\), and \(x+2\). If \(x+2\) is the longest side, what is the range of possible values for \(x\)?

What is the solution set to the equation \(\sqrt{9x^2 - 54x + 81} = 3x - 9\)?

What is the sum of the solutions to the equation \(\sin^2 x - \sin x - 6 = 0\) over the interval \([0, 2\pi]\)?

The function \(f(x) = \frac{2x}{x^2 + 1}\) reaches its maximum on the interval \([-1, 2]\) at \(x =\):

A geometric series has a first term of 5 and a common ratio of \(\frac{1}{2}\). What is the sum of the first 6 terms?

If \( \log_2 (8x) = 6 \), what is \(x\)?

The roots of the equation \(x^2 - 8x + 15 = 0\) are:

If the points (1,1), (4,y), and (9,9) lie on the graph of a quadratic function \(y = ax^2 + bx + c\), what is the value of \(y\) when \(x = 4\)?

For the circle with equation \((x - 4)^2 + (y + 3)^2 = 16\), what are the coordinates of its center and the length of its radius?

What is the area of a triangle with vertices at \((0,0)\), \((4,0)\), and \((0,3)\)?

Solve for \(x\) in the equation \(4^{x+1} = 16^x\).

If the function \( f(x) = \frac{x^2 - 1}{x - 1} \) is evaluated for \( x \neq 1 \), which expression is equivalent to \( f(x) \)?

What is the sum of the solutions to the equation \( 2x^2 - 8x + 8 = 0 \)?

A circle has equation \( x^2 + y^2 + 6x - 4y + 9 = 0 \). What is the radius of this circle?

If \( \log_b(3) = a \) and \( \log_b(5) = c \), what is \( \log_b(45) \) in terms of \( a \) and \( c \)?

For \( f(x) = \sqrt{2x + 3} \), what is the domain of \( f \)?

If \( i \) is the imaginary unit, what is the product \( (3 - 4i)(3 + 4i) \)?

How many integer solutions satisfy the inequality \( |x - 3| > 2 \)?

The equation \( e^{2x} - 7e^x + 10 = 0 \) can be rewritten as a quadratic equation. What are the roots?

A cubic function \( f(x) = x^3 - 6x^2 + 9x \) has a factor \( x \). What are the other factors?

Given the function \( f(x) = \frac{x^2 - 4x + 4}{x - 2} \), what is the value of \( f(2) \)?

What is the vertex of the parabola represented by the equation \( y = 3(x - 2)^2 - 5 \)?

If \( f(x) = \frac{1}{x-1} \) and \( g(x) = \frac{1}{x+1} \), what is \( f(g(1)) \)?

What is the remainder when the polynomial \( x^3 - 4x^2 + x + 6 \) is divided by \( x - 2 \)?

The function \( f(x) = x^2 - 6x + 8 \) reaches its minimum at \( x = \)?

What is the inverse of the function \( f(x) = 3x - 7 \)?

If \( \tan(\theta) = 3 \) and \( \theta \) is in the first quadrant, what is \( \sin(\theta) \)?

How many real roots does the equation \( x^4 - 16x^2 + 64 = 0 \) have?