Function properties and transformations

A change outside the function affects y: f(x) + a shifts up by a and f(x) − a shifts down. A change in the argument acts horizontally: f(x − a) shifts right by a and f(x + a) left, for a > 0.

y = −f(x) reverses every y sign and reflects the graph in the x-axis. y = af(x) multiplies each y by a; for a > 1 it stretches vertically by a, and for 0 < a < 1 it compresses.

The inner function value must lie in the outer function domain. Form a non-negative radicand inequality and solve using distance.

Evaluate the inner function first, then substitute its value into the outer function.

The logarithm argument must be positive, and the denominator must be nonzero. Apply both conditions. A zero logarithm value is allowed if the denominator is nonzero.

A change outside the function affects y: f(x) + a shifts up by a and f(x) − a shifts down. A change in the argument acts horizontally: f(x − a) shifts right by a and f(x + a) left, for a > 0.

Check which input of f the given x produces. Use the known f value, then apply the outer operations.

Substitute g(x) for the whole argument of f. Expand and solve the resulting linear equation.

Determine monotonicity from the base. Evaluate the interval endpoints, accounting for both shifts. Include the endpoint values because the interval is closed.

Form g(f(x)), multiplying the entire expression of f by the coefficient of g. Collect the x terms and constant terms separately.