Reciprocal and square of functions (HL)

This section is not frequently tested. Students should be comfortable reasoning out the answers as opposed to memorizing the effects.

On a calculator, reciprocal of function is f(x)⁻¹, and square of function is f(x)². How not to enter powers of functions on TI-84 Plus

Contents

Reciprocal of functions

See also ratio of quadratic and linear functions

This is about f(x)−1=1f(x)f(x)^{-1} = \frac{1}{f(x)}. Not to be confused with inverse functions, which are often represented by f−1(x)f^{-1}(x) or simply f−1f^{-1}.

This table summarizes how points on ff are transformed into f(x)−1f(x)^{-1}.

f(x)f(x)f(x)−1f(x)^{-1}
zerosvertical asymptotes
vertical asymptotesholes on xx-axis
where 0<y<10 < y < 1where y>1y > 1
where y>1y > 1where 0<y<10 < y < 1
where −1<y<0-1 < y < 0where y<−1y < -1
where y<−1y < -1where −1<y<0-1 < y < 0
where y=±1y = \pm1not transformed (static points)
horizontal asymptote y=ky = khorizontal asymptote y=1ky = \frac1k
horizontal asymptote y=0y = 0end behavior ±∞\pm\infty
end behavior ±∞\pm\inftyhorizontal asymptote y=0y = 0

Notably, f(x)−1f(x)^{-1} has holes (a single point outside the domain) when there would otherwise be zeros.

For all xx in the domain of ff such that f(x)≠0f(x) \neq 0

f(x)⋅f(x)−1=1,    f(x)≠0f(x) \cdot f(x)^{-1} = 1, \;\; f(x) \neq 0

The reciprocal of an odd or even function is also odd or even.

How to enter reciprocal functions on TI-84 Plus

Square of functions

This table summarizes how points on ff are transformed into f(x)2f(x)^{2}.

f(x)f(x)f(x)2f(x)^{2}
zeroszeros (static points)
where 0<y<10 < y < 1 or −1<y<0-1< y < 0where 0<y<10 < y < 1
where y>1y > 1 or y<−1y < -1where y>1y > 1
where y=1y = 1not transformed (static points)
where y=−1y = -1reflected to where y=1y = 1.
horizontal asymptote y=ky = khorizontal asymptote y=k2y = k^2
vertical asymptotesvertical asymptotes but both sides approach +∞+\infty
end behavior ±∞\pm\inftyend behavior +∞+\infty

How not to enter square functions on TI-84 Plus