Pre Calculus Unit 3 Practice Quiz

The square root function is only defined for nonnegative arguments, so the domain is x ≥ 0. The other options fail to capture the requirement for nonnegative x values.

Adding 3 to the function values shifts the graph vertically upward by 3. This transformation does not affect the horizontal alignment or scale of the graph.

The composition (f ∘ g)(x) is computed by substituting g(x) into f, resulting in f(g(x)) = 2*(x + 5) = 2x + 10. The other options do not correctly follow the order of operations required for composition.

To find the inverse function, you swap x and y in the equation y = x + 2 and solve for y, which gives y = x - 2. This confirms that f❻¹(x) = x - 2 is the correct inverse.

A linear function has the form f(x) = mx + b. The function 3x - 4 fits this form while the other options represent quadratic, radical, and rational functions.

The y-intercept is found by evaluating the function at x = 0. For f(x) = 2x² - 3x + 1, f(0) equals 1, so the y-intercept is 1.

Since 27 can be written as 3³, equating 3^x to 3³ shows that x must equal 3. This makes 3 the unique solution based on properties of exponents.

The x-coordinate of the vertex of a quadratic function is given by -b/(2a). This formula is derived from either completing the square or using calculus to find the function's extreme value.

Multiplying a function by -1 reflects its graph over the x-axis, changing the sign of all its y-values. This is a standard transformation in function analysis.

The composition (f ∘ g)(x) is obtained by substituting g(x) into f(x), which gives (x + 1)². This correctly represents the composite function before any expansion.

For the logarithmic function log(x - 2) to be defined, the argument x - 2 must be positive. Therefore, x must be greater than 2.

An exponential growth function features a base greater than 1; here, 2^x grows exponentially as x increases. Multiplying by 3 does not change the growth behavior.

The function is undefined when the denominator is zero. Since x² - 9 factors into (x - 3)(x + 3), x cannot be 3 or -3.

The composition (f ∘ g)(x) is f(g(x)) which equals (√x)². This simplifies to x, assuming x is nonnegative in accordance with the domain of √x.

Substituting x = 5 into f(x) = |x - 3| gives |5 - 3|, which equals 2. This direct evaluation confirms that 2 is the correct answer.

To find the inverse, swap x and y in the equation y = (2x + 3)/(x - 1) and solve for y. This process yields f❻¹(x) = (x + 3)/(x - 2), which is the correct inverse function.

By substituting u = x², the equation transforms to u² - 5u + 4 = 0, which factors as (u - 1)(u - 4) = 0. Solving for u gives u = 1 and u = 4, leading to x = ±1 and x = ±2, a total of four real solutions.

The polynomial factors neatly into (x - 1)(x - 2)(x - 3), which can be verified by either synthetic division or the Rational Root Theorem. This factorization demonstrates that the roots are 1, 2, and 3.

For rational functions where the highest power in the numerator and denominator is the same, the limit as x approaches infinity is the ratio of the leading coefficients. Thus, the limit is 3/5.

Rewriting the equation gives sin²(x) = 1/2, and taking the square root results in sin(x) = ±√2/2. This leads to four solutions in the interval [0, 2π): π/4, 3π/4, 5π/4, and 7π/4.