I now have sufficient content from Guyton and Hall. Here is the 2-mark short answer:
The cell membrane (plasma membrane) is a thin, pliable, elastic lipid bilayer interspersed with proteins, approximately 7.5 to 10 nm thick. Its major functions are:
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Selective barrier / permeability control - The lipid bilayer is impermeable to water-soluble substances (ions, glucose, urea) but freely permeable to fat-soluble substances (O₂, CO₂, alcohol), thereby regulating what enters and exits the cell.
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Transport of substances - Integral membrane proteins form channels and carrier systems that mediate diffusion, facilitated diffusion, and active transport of ions, nutrients, and waste products across the membrane.
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Cell signaling and receptor function - Membrane proteins act as receptors for hormones and neurotransmitters, initiating intracellular signal cascades (e.g., G-protein-coupled pathways, enzyme-linked receptors).
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Enzymatic activity - Peripheral proteins attached to the membrane surface function as enzymes catalyzing chemical reactions.
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Cell recognition and adhesion - Membrane carbohydrates form the glycocalyx, which serves as a recognition layer for cell-to-cell interactions and adhesion sites for extracellular proteins.
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Endocytosis and exocytosis - The membrane mediates ingestion of large particles (phagocytosis) and fluid/solutes (pinocytosis), as well as secretion of substances out of the cell.
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Maintenance of electrochemical gradients - Via the Na⁺-K⁺ ATPase pump embedded in the membrane, a negative intracellular electrical voltage is maintained, which is fundamental to nerve and muscle function.