From The Collaborative International Dictionary of English v.0.48:
Undertake \Un`der*take"\, v. t. [imp. Undertook; p. p. Undertaken; p. pr. & vb. n. Undertaking.] [Under + take.] 1. To take upon one's self; to engage in; to enter upon; to take in hand; to begin to perform; to set about; to attempt. [1913 Webster]
To second, or oppose, or undertake The perilous attempt. --Milton. [1913 Webster]
2. Specifically, to take upon one's self solemnly or expressly; to lay one's self under obligation, or to enter into stipulations, to perform or to execute; to covenant; to contract. [1913 Webster]
I 'll undertake to land them on our coast. --Shak. [1913 Webster]
3. Hence, to guarantee; to promise; to affirm. [1913 Webster]
And he was not right fat, I undertake. --Dryden. [1913 Webster]
And those two counties I will undertake Your grace shall well and quietly enjoiy. --Shak. [1913 Webster]
I dare undertake they will not lose their labor. --Woodward. [1913 Webster]
4. To assume, as a character. [Obs.] --Shak. [1913 Webster]
5. To engage with; to attack. [Obs.] [1913 Webster]
It is not fit your lordship should undertake every companion that you give offense to. --Shak. [1913 Webster]
6. To have knowledge of; to hear. [Obs.] --Spenser. [1913 Webster]
7. To take or have the charge of. [Obs.] "Who undertakes you to your end." --Shak. [1913 Webster]
Keep well those that ye undertake. --Chaucer. [1913 Webster]
From The Collaborative International Dictionary of English v.0.48:
Undertaking \Un`der*tak"ing\, n. 1. The act of one who undertakes, or engages in, any project or business. --Hakluyt. [1913 Webster]
2. That which is undertaken; any business, work, or project which a person engages in, or attempts to perform; an enterprise. [1913 Webster]
3. Specifically, the business of an undertaker, or the management of funerals. [1913 Webster]
4. A promise or pledge; a guarantee. --A. Trollope. [1913 Webster]