Free TCE Mathematics Methods Flashcard Review

The output of the function when its input is a. It is found by substituting a into the rule, provided a belongs to the function’s domain.

The domain is the set of allowed inputs. The range is the set of outputs actually attained from those inputs; it is not automatically the same as the stated codomain.

The extremal output may occur at a different, allowed input. For x² on (−1,2], the minimum 0 occurs at interior input 0 and the maximum 4 occurs at 2, so the range is [0,4].

Every input making the original denominator 0 is excluded. Cancellation may simplify the rule but does not restore an input where the original expression was undefined.

Its argument must be strictly positive. Zero is excluded as well as negative values, so logarithm domain inequalities are strict.

The radicand must be strictly positive, because a zero radicand would also make the denominator 0. A square root used only in a numerator may allow a zero radicand.

Yes. A function requires one output per input, not one input per output. Distinct inputs sharing an output instead mean the function is not one-to-one.

Zeros are allowed inputs where f(x)=0, giving x-axis intersections. The y-intercept is (0, f(0)) only if 0 belongs to the domain.

It is [1/4,1]. The reciprocal decreases on the positive interval, so the smaller output comes from the larger input, and both endpoints are attained.

Take their intersection: an input must satisfy every restriction at once. For √(x−1)+ln(4−x), the conditions x≥1 and x<4 combine to [1,4).

It means x−a is a factor of the polynomial P. Conversely, if x−a divides P exactly, then substituting a gives 0.

The remainder is P(a). This lets an unknown coefficient be constrained by a value equation without carrying out full polynomial division.

The graph normally touches the axis and returns to the same side. A factor (x−a)² does not change sign across a, though other factors must be checked for the local graph direction.

An odd power of x−a changes sign across a. The graph crosses the axis; higher odd multiplicity can make the crossing locally flat.

It is C(5,2)=10. The coefficient counts the ways to choose the x term from exactly two of the five brackets.

The term containing xʳ has the factor (−1)ʳ. Even r gives a positive sign and odd r a negative sign, with binomial coefficients supplying the magnitudes.

Set u=x², solve the resulting quadratic in u, and retain only u≥0 for real x. Each positive u gives two roots ±√u, while u=0 gives one distinct root.

For large |x|, the highest-power term dominates. Its degree parity and coefficient sign determine whether the two ends go in the same or opposite directions.

Include every valid zero, including an isolated even-multiplicity root outside a positive-sign interval. Strict inequalities exclude all equality roots.

Multiply the required root factors with their multiplicities, attach a scale coefficient, and use the nonzero point value to determine that scale. Check all supplied conditions afterwards.