1001 Algebra Problems.PDF

c.Note that,x+ 21 = 10x (^12)  or equivalently,x– 10 x (^12)  21 = 0 can be written as (x (^12)  )^2 – 10(x (^12)  ) ...

ANSWERS & EXPLANATIONS– b.We must first rewrite the equation in a nicer form. Observe that 3 +x 41  x 21  = 0 x ...

b.Let u=x^2 +x. Observe that (x^2 +x)^2 + 12 = 8(x^2 +x)or equivalently (x^2 +x)^2 – 8(x^2 +x) + 12 = 0, can be written as u^2 ...

b.Observe that 6x– 13 ^4 x+ 6 = 0 can be written as 6(^4 x)^2 – 13(^4 x) + 6 = 0. Let u = ^4 x.Rewriting the original ...

b.The domain of the function is the set of all real numbers, so any real number can be sub- stituted for x. The range of a func ...

b. = = = – 39 ((xx++ 11 )) 2 = –3(x+ 1) = –g(x) d.Since the function 2g(x)h(x) = is a rational function, its domain is the ...

into this equation. There are no x-values that will generate an undefined or imaginary y value. However, it is impossible to gen ...

f(x) = –(x– 1)^2 + 3 = –(x^2 – 2x+ 1) + 3 = x^2 + 2x– 1 + 3 = x^2 – 2x+ 2 Finally, simplify the original expression f(x+ h) – ...

Set 45 (Page 107) b.The radicand of an even-indexed radical term (e.g., a square root) must be nonnegative if in the numerator ...

c.The minimum value for both functions occurs at their vertex, which occurs at (0, 0). Also, for any positive real number a, th ...

b.If there is no x-value that satisfies the equa- tion,f(x) = 0 then the graph ofy= f(x) does not cross the horizontal line y= ...

b.First, note that x^2 + 1 does not factor, so x= 1 is a vertical asymptote for the graph off. Since the degree of the numerato ...

ANSWERS & EXPLANATIONS– d.The y-values on all three graphs increase as the x-values move from left to right in the interval ...

f(x) = x^3 g(x) = –1 – x^2 –10 –8 –6 –4 –2^246810 –2 –4 –6 –8 –10 10 8 6 4 2 –10 –8 –6 –4 –2 2 4 6 8 10 –2 –4 –6 –8 –10 10 8 6 4 ...

d.Statement ais true because, by definition, a positive slope ofmmeans that the graph of the line rises vertically m units for ...

b.The graph ofy= g(x+ h) + kis obtained by shifting the graph ofy= g(x) right (resp. left) hunits ifh< 0 (resp.h> 0), and ...

b.(ex+ e–x)^2 = (ex)^2 + 2(ex)(e–x) + (e–x)^2 = e^2 x + 2 + e–2x= e^2 x+ e–2x+ 2 d. = = = 5^8 x–4= (5^4 )^2 x–1 = 625^2 x–1 ...

c.The graph off(x) = –^34  x is always below the x-axis and increases as xmoves from left to right, as shown by its graph: ...

d.The graph off(x) = –e2–x– 3 can be obtained by reflecting the graph ofg(x) = e–x about the y-axis, then shifting it to the le ...

c.We rewrite both sides of the equation as a power of 2, then equate the exponents and solve for x: 4 x+1= (^12 )^2 x (2^2 ) ...