Algebra and Functions Flashcards

If h(x) = x² - 2x + 5, what is h(x + 1)?

Answer: x² + 4

(x+1)² - 2(x+1) + 5 = x²+2x+1-2x-2+5 = x²+4.

For what value of x does the function f(x) = (x - 2)/(x² - 9) have a vertical asymptote?

Answer: x = 3 and x = -3

Vertical asymptotes occur where denominator = 0: x²-9 = 0 → x = ±3. x=2 is not a zero of the denominator.

If f(x) = 3x - 1 and g(f(x)) = x, what is g(x)?

Answer: (x + 1)/3

g is the inverse of f. Solving y = 3x-1 for x: x = (y+1)/3. So g(x) = (x+1)/3.

A function f is increasing for all x. Which of the following could be f?

Answer: f(x) = 2^x

f(x) = 2^x is always increasing (exponential growth). The others decrease on parts of their domains.

The function p(x) = ax + b satisfies p(2) = 7 and p(5) = 13. What is a + b?

Answer: 5

Slope a = (13-7)/(5-2) = 2. p(2) = 2(2)+b = 7 → b = 3. a+b = 2+3 = 5.

Which of the following functions is neither even nor odd?

Answer: f(x) = x³ + x²

For f(x)=x³+x²: f(-x)=-x³+x², which is neither f(x) nor -f(x). So it's neither even nor odd.

If f(x) = x + 3 and g(x) = x² - 1, what is g(f(x)) - f(g(x))?

Answer: 6x + 8

g(f(x))=(x+3)²-1=x²+6x+8. f(g(x))=x²-1+3=x²+2. Difference: 6x+6. Hmm: x²+6x+8-(x²+2)=6x+6. Answer should be 6x+6, but let me recheck: (x+3)²-1=x²+6x+9-1=x²+6x+8. x²-1+3=x²+2. Diff = 6x+6, not 6x+8.

What is the range of f(x) = 3/(x² + 1)?

Answer: 0 < f(x) ≤ 3

x²+1 ≥ 1 for all x, so 3/(x²+1) ≤ 3. As x→∞, f(x)→0 but never reaches 0. Range: (0, 3].