The Federal Reserve Board announced yesterday another sharp boost in interest rates here at home, primarily to bolster the dollar, which has been battered this week in the face of European uncertainty about President Carter's round of Cabinet firings.

The move came as the Commerce Department reported that the output of the nation's economy declined last quarter at a sharp 3.3 percent annual rate, signalling the start of the predicted recession. At the same time inflation speeded to a 9.9 percent pace from 9.3 percent.

The Fed's move in raising interest rates, considered highly unusual at the start of a recession, underscored Carter's dilemma in managing the economy -- although the nation clearly is headed into a slump, inflation and the dollar problem are preventing any quick remedies.

The Fed took its action mainly by raising the discount rate -- the interest it charges on loans to member banks -- to a record 10 percent, from 9.5 percent. It also nudged up other interest rates, boosting its key federal funds rate to 10.5 percent.

The dollar recovered a bit in the wake of the early-morning announcement, but Wall Street analysts said the Fed still had to intervene in the markets with heavy buying of dollars to turn the situation around. Some economists fear the rise in interest rates could worsen the recession.

The decline in the nation's output reflected in a report on the "real" gross national product, the actual volume of goods and services that the nation produces, marked the first actual drop since early 1975 at the bottom of the last recession.

Courtenay M. Slater, the Commerce Department's chief economist, told reporters at a briefing that the administration expects output to decline again in the current quarter, but reiterated earlier predictions that the recession "will not be deep and will not be long."

The White House has predicted two quarters of decline in "real" putput, followed by a relatively weak recovery later this year and in 1980. Private economists are sligthly more pessimistic, but they still expect the slump to be relatively brief.

The Fed's action yesterday was supported by the Carter administration Ousted Treasury Secretary W. Michael Blumenthal, who will stay on the job temporarily after having been fired on Thursday, issued a statment endorsing the Fed moves "on behalf of the president."

The Fed's decision to boost interest rates was prompted by two developments: Recent interest-rate increases by European nations have drawn investors away from U.S. dollars, devaluing the American currency, and the dollar has fallen further in the wake of Carter's Cabinet shakeup.

Blumenthal said yesterday that the Fed's action -- the first increase in the discount rate since last November's dollar-rescue effort -- would be "helpful in dealing with speculative pressure on the dollar." Nevertheless, the Fed again had to intervene in the markets to bolster the dollar.

The second-quarter decline in output was concentrated mainly in the automobile sector, which has been hit hard by the gasoline shortage. However, the weakness showed throughout the economy. And business accumulated more excess inventories, which could delay any resumption in economic growth.

The department also reported that real final sales, the government broadest measure of overall buying activity, declined at a sharp annual rate of 4.5 percent. Consumer spending fell at a 3.6 percent pace. And business spending on new plant and equipment pluged at a 5.1 percent rate.

There also was a significant deterioration in the U.S. underlying trade position. Net exports declined at a $7 billion annual rate during the quarter, reflecting both a slowdown in overall exports and a sharp increase in imports, partly a result of higher oil prices.

The statistics on inflation were no more encouraging. The department's GNP price index, the most comprehensive measure of inflation, showed prices rose at a 9.9 percent, annual rate in the April-June period, compared with a 9.3 percent pace in the previous quarter.

The figures also showed the first decline in four years in the real income of Americans. Disposable personnal income, adjusted for inflation, fell at 1.7 percent annual rate last quarter after rising at a 2.1 percent pace in the previous period.

For all the decline last quarter, the administration and most private economists still are urging a wait-and-see attitude before pushing for any tax cut. Slater told the briefying yesterday policymakers must "watch developments carefully and be prepared" to act later on.

Yesterday's report also contained the department's annunal revisions of GNP statistics for the previous three years. The figures showed real output rose at a 1.1 percent pace in the first quarter of this year and at a 5.6 percent pace in the final three months of 1978. CAPTION: Graph, FEDERAL RESERVE DISCOUNT RATE, Source: Federal Reserve Board